asciimath/TeX Samples.txt
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<link type="text/css" rel="stylesheet" href="TeX%20Samples%20%7C_files/widget.css"><style type="text/css">#twtr-widget-1 .twtr-popular { display: inline-block; }</style><style type="text/css">#twtr-widget-1 .twtr-avatar { display: block; } #twtr-widget-1 .twtr-user { display: inline; } #twtr-widget-1 .twtr-tweet-text { margin-left: 40px; }</style><style type="text/css">#twtr-widget-1 em { display: block; }</style><style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
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@font-face {font-family: MathJax_Main; src: url('http://cdn.mathjax.org/mathjax/latest/fonts/HTML-CSS/TeX/woff/MathJax_Main-Italic.woff') format('woff'), url('http://cdn.mathjax.org/mathjax/latest/fonts/HTML-CSS/TeX/otf/MathJax_Main-Italic.otf') format('opentype'); font-style: italic}
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border-right: 1px dashed #BBBBBB !important;
border-bottom: 1px solid #000000 !important;
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top: 0 !important;
left: -15px !important;
width: 14px !important;
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border-left: 1px solid #BBBBBB !important;
border-right: 1px solid #000000 !important;
border-bottom: 1px dashed #BBBBBB !important;
}
.overflowRulerX > .firebugRulerV {
left: 0 !important;
}
.overflowRulerY > .firebugRulerH {
top: 0 !important;
}
/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
.fbProxyElement {
position: fixed !important;
pointer-events: auto !important;
}
</style></head>
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<h1 class="entry-title">TeX Samples</h1>
<div class="entry-content">
<p>The following equations are represented in the HTML source code as LaTeX expressions.</p>
<div class="math-header">The Lorenz Equations</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-1-Frame" class="MathJax"><nobr><span id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 136px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(-0.085em, 1000em, 4.111em, -0.297em); top: -2.263em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span style="padding-right: 0.167em; padding-left: 0.167em;" id="MathJax-Span-3" class="mtable"><span style="display: inline-block; position: relative; width: 7.013em; height: 0px;"><span style="position: absolute; clip: rect(2.335em, 1000em, 6.234em, -0.464em); top: -4.58em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.593em; height: 0px;"><span style="position: absolute; clip: rect(3.129em, 1000em, 4.16em, -0.45em); top: -5.373em; right: 0em;"><span id="MathJax-Span-4" class="mtd"><span id="MathJax-Span-5" class="mrow"><span id="MathJax-Span-6" class="texatom"><span id="MathJax-Span-7" class="mrow"><span id="MathJax-Span-8" class="munderover"><span style="display: inline-block; position: relative; width: 0.593em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.45em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-9" class="mi">x</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(1.863em, 1000em, 2.307em, -0.295em); top: -2.722em; left: 0.082em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-10" class="mo">˙</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.129em, 1000em, 4.354em, -0.464em); top: -3.927em; right: 0em;"><span id="MathJax-Span-20" class="mtd"><span id="MathJax-Span-21" class="mrow"><span id="MathJax-Span-22" class="texatom"><span id="MathJax-Span-23" class="mrow"><span id="MathJax-Span-24" class="munderover"><span style="display: inline-block; position: relative; width: 0.569em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.845em, -0.464em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-25" class="mi">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(1.863em, 1000em, 2.307em, -0.295em); top: -2.722em; left: 0.085em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-26" class="mo">˙</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.129em, 1000em, 4.16em, -0.45em); top: -2.506em; right: 0em;"><span id="MathJax-Span-37" class="mtd"><span id="MathJax-Span-38" class="mrow"><span id="MathJax-Span-39" class="texatom"><span id="MathJax-Span-40" class="mrow"><span id="MathJax-Span-41" class="munderover"><span style="display: inline-block; position: relative; width: 0.567em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.45em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-42" class="mi">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(1.863em, 1000em, 2.307em, -0.295em); top: -2.722em; left: 0.083em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-43" class="mo">˙</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.58em;"></span></span><span style="position: absolute; clip: rect(2.551em, 1000em, 6.697em, -0.429em); top: -4.849em; left: 0.871em;"><span style="display: inline-block; position: relative; width: 6.142em; height: 0px;"><span style="position: absolute; clip: rect(1.782em, 1000em, 3.106em, -0.429em); top: -4.08em; left: 0em;"><span id="MathJax-Span-11" class="mtd"><span id="MathJax-Span-12" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-13" class="mo">=</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;" id="MathJax-Span-14" class="mi">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-15" class="mo">(</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-16" class="mi">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-17" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-18" class="mi">x</span><span style="font-family: MathJax_Main;" id="MathJax-Span-19" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(2.09em, 1000em, 3.072em, -0.429em); top: -2.634em; left: 0em;"><span id="MathJax-Span-27" class="mtd"><span id="MathJax-Span-28" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-29" class="mo">=</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.278em;" id="MathJax-Span-30" class="mi">ρ</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-31" class="mi">x</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-32" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-33" class="mi">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-34" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-35" class="mi">x</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-36" class="mi">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.827em, 1000em, 3.061em, -0.429em); top: -1.213em; left: 0em;"><span id="MathJax-Span-44" class="mtd"><span id="MathJax-Span-45" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-46" class="mo">=</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-47" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-48" class="mi">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.007em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-49" class="mi">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-50" class="mo">+</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-51" class="mi">x</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-52" class="mi">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.006em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.849em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 4.617em; vertical-align: -2.019em;"></span></span></nobr></span></div><script id="MathJax-Element-1" type="math/tex; mode=display">\begin{aligned}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{aligned} </script>
<div class="math-header">The Cauchy-Schwarz Inequality</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-2-Frame" class="MathJax"><nobr><span id="MathJax-Span-53" class="math"><span style="display: inline-block; position: relative; width: 283px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(2.417em, 1000em, 5.937em, -0.248em); top: -4.526em; left: 0em;"><span id="MathJax-Span-54" class="mrow"><span id="MathJax-Span-55" class="msubsup"><span style="display: inline-block; position: relative; width: 5.571em; height: 0px;"><span style="position: absolute; clip: rect(2.615em, 1000em, 5.937em, -0.248em); top: -4.526em; left: 0em;"><span id="MathJax-Span-56" class="mfenced"><span style="vertical-align: 0em;" id="MathJax-Span-75" class="mo"><span style="font-family: MathJax_Size4;">(</span></span><span id="MathJax-Span-58" class="mrow"><span id="MathJax-Span-59" class="munderover"><span style="display: inline-block; position: relative; width: 1.455em; height: 0px;"><span style="position: absolute; clip: rect(2.66em, 1000em, 4.383em, -0.43em); top: -3.772em; left: 0em;"><span style="font-family: MathJax_Size2; vertical-align: 0em;" id="MathJax-Span-60" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.772em;"></span></span><span style="position: absolute; clip: rect(1.718em, 1000em, 2.64em, -0.451em); top: -1.263em; left: 0.081em;"><span id="MathJax-Span-62" class="texatom"><span id="MathJax-Span-63" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-64" class="mi">k</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-65" class="mo">=</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-66" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(1.743em, 1000em, 2.486em, -0.47em); top: -3.467em; left: 0.512em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-67" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span><span style="padding-left: 0.167em;" id="MathJax-Span-68" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(1.876em, 1000em, 2.65em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-69" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.167em; left: 0.539em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-70" class="mi">k</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span><span id="MathJax-Span-71" class="msubsup"><span style="display: inline-block; position: relative; width: 0.883em; height: 0px;"><span style="position: absolute; clip: rect(1.623em, 1000em, 2.651em, -0.445em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-72" class="mi">b</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.167em; left: 0.431em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-73" class="mi">k</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span></span><span style="vertical-align: 0em;" id="MathJax-Span-76" class="mo"><span style="font-family: MathJax_Size4;">)</span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.75em; vertical-align: 0em;"></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.75em; vertical-align: 0em;"></span></span><span style="display: inline-block; width: 0px; height: 4.526em;"></span></span><span style="position: absolute; top: -3.847em; left: 5.119em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-77" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-78" class="mo">≤</span><span style="padding-left: 0.278em;" id="MathJax-Span-79" class="mfenced"><span style="vertical-align: 0em;" id="MathJax-Span-80" class="mo"><span style="font-family: MathJax_Size4;">(</span></span><span id="MathJax-Span-81" class="mrow"><span id="MathJax-Span-82" class="munderover"><span style="display: inline-block; position: relative; width: 1.455em; height: 0px;"><span style="position: absolute; clip: rect(2.66em, 1000em, 4.383em, -0.43em); top: -3.772em; left: 0em;"><span style="font-family: MathJax_Size2; vertical-align: 0em;" id="MathJax-Span-83" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.772em;"></span></span><span style="position: absolute; clip: rect(1.718em, 1000em, 2.64em, -0.451em); top: -1.263em; left: 0.081em;"><span id="MathJax-Span-84" class="texatom"><span id="MathJax-Span-85" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-86" class="mi">k</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-87" class="mo">=</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-88" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(1.743em, 1000em, 2.486em, -0.47em); top: -3.467em; left: 0.512em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-89" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span><span style="padding-left: 0.167em;" id="MathJax-Span-90" class="msubsup"><span style="display: inline-block; position: relative; width: 0.991em; height: 0px;"><span style="position: absolute; clip: rect(1.876em, 1000em, 2.65em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-91" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(1.738em, 1000em, 2.532em, -0.45em); top: -2.715em; left: 0.539em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-92" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(1.664em, 1000em, 2.486em, -0.451em); top: -1.991em; left: 0.539em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-93" class="mi">k</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span></span><span style="vertical-align: 0em;" id="MathJax-Span-94" class="mo"><span style="font-family: MathJax_Size4;">)</span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.75em; vertical-align: 0em;"></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.75em; vertical-align: 0em;"></span></span><span id="MathJax-Span-95" class="mfenced"><span style="vertical-align: 0em;" id="MathJax-Span-96" class="mo"><span style="font-family: MathJax_Size4;">(</span></span><span id="MathJax-Span-97" class="mrow"><span id="MathJax-Span-98" class="munderover"><span style="display: inline-block; position: relative; width: 1.455em; height: 0px;"><span style="position: absolute; clip: rect(2.66em, 1000em, 4.383em, -0.43em); top: -3.772em; left: 0em;"><span style="font-family: MathJax_Size2; vertical-align: 0em;" id="MathJax-Span-99" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.772em;"></span></span><span style="position: absolute; clip: rect(1.718em, 1000em, 2.64em, -0.451em); top: -1.263em; left: 0.081em;"><span id="MathJax-Span-100" class="texatom"><span id="MathJax-Span-101" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-102" class="mi">k</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-103" class="mo">=</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-104" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(1.743em, 1000em, 2.486em, -0.47em); top: -3.467em; left: 0.512em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-105" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span><span style="padding-left: 0.167em;" id="MathJax-Span-106" class="msubsup"><span style="display: inline-block; position: relative; width: 0.883em; height: 0px;"><span style="position: absolute; clip: rect(1.623em, 1000em, 2.651em, -0.445em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-107" class="mi">b</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(1.738em, 1000em, 2.532em, -0.45em); top: -2.792em; left: 0.431em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-108" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(1.664em, 1000em, 2.486em, -0.451em); top: -2.067em; left: 0.431em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-109" class="mi">k</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span></span><span style="vertical-align: 0em;" id="MathJax-Span-110" class="mo"><span style="font-family: MathJax_Size4;">)</span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.75em; vertical-align: 0em;"></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.75em; vertical-align: 0em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.526em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 3.833em; vertical-align: -1.512em;"></span></span></nobr></span></div><script id="MathJax-Element-2" type="math/tex; mode=display"> \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) </script>
<div class="math-header">A Cross Product Formula</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-3-Frame" class="MathJax"><nobr><span id="MathJax-Span-111" class="math"><span style="display: inline-block; position: relative; width: 224px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(2.296em, 1000em, 8.626em, -0.46em); top: -5.711em; left: 0em;"><span id="MathJax-Span-112" class="mrow"><span id="MathJax-Span-113" class="msubsup"><span style="display: inline-block; position: relative; width: 1.314em; height: 0px;"><span style="position: absolute; clip: rect(1.9em, 1000em, 2.916em, -0.46em); top: -2.748em; left: 0em;"><span id="MathJax-Span-114" class="texatom"><span id="MathJax-Span-115" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-116" class="mi">V</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -2.221em; left: 0.862em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-117" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-118" class="mo">×</span><span style="padding-left: 0.222em;" id="MathJax-Span-119" class="msubsup"><span style="display: inline-block; position: relative; width: 1.314em; height: 0px;"><span style="position: absolute; clip: rect(1.9em, 1000em, 2.916em, -0.46em); top: -2.748em; left: 0em;"><span id="MathJax-Span-120" class="texatom"><span id="MathJax-Span-121" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-122" class="mi">V</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -2.221em; left: 0.862em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-123" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-124" class="mo">=</span><span style="padding-left: 0.278em;" id="MathJax-Span-125" class="mfenced"><span style="vertical-align: 3.253em;" id="MathJax-Span-126" class="mo"><span style="display: inline-block; position: relative; width: 0.278em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.237em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 1.77em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.403em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.568em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.733em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.101em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.936em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="padding-right: 0.167em; padding-left: 0.167em;" id="MathJax-Span-127" class="mtable"><span style="display: inline-block; position: relative; width: 6.061em; height: 0px;"><span style="position: absolute; clip: rect(3.587em, 1000em, 9.811em, -0.445em); top: -6.897em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.761em; height: 0px;"><span style="position: absolute; clip: rect(1.891em, 1000em, 2.909em, -0.445em); top: -5.201em; left: 50%; margin-left: -0.162em;"><span id="MathJax-Span-128" class="mtd"><span id="MathJax-Span-129" class="mrow"><span id="MathJax-Span-130" class="texatom"><span id="MathJax-Span-131" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-132" class="mi">i</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; clip: rect(2.434em, 1000em, 4.86em, -0.365em); top: -4.449em; left: 50%; margin-left: -0.88em;"><span id="MathJax-Span-143" class="mtd"><span id="MathJax-Span-144" class="mrow"><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-145" class="mfrac"><span style="display: inline-block; position: relative; width: 1.521em; height: 0px;"><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -3.37em; left: 50%; margin-left: -0.7em;"><span id="MathJax-Span-146" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-147" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-148" class="mi">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -2.004em; left: 50%; margin-left: -0.593em;"><span id="MathJax-Span-149" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-150" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-151" class="mi">u</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.521em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(2.434em, 1000em, 4.86em, -0.365em); top: -1.946em; left: 50%; margin-left: -0.88em;"><span id="MathJax-Span-164" class="mtd"><span id="MathJax-Span-165" class="mrow"><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-166" class="mfrac"><span style="display: inline-block; position: relative; width: 1.521em; height: 0px;"><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -3.37em; left: 50%; margin-left: -0.7em;"><span id="MathJax-Span-167" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-168" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-169" class="mi">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.024em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -2.004em; left: 50%; margin-left: -0.539em;"><span id="MathJax-Span-170" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-171" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-172" class="mi">v</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.521em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.897em;"></span></span><span style="position: absolute; clip: rect(3.587em, 1000em, 9.811em, -0.556em); top: -6.897em; left: 2.761em;"><span style="display: inline-block; position: relative; width: 1.707em; height: 0px;"><span style="position: absolute; clip: rect(1.891em, 1000em, 3.109em, -0.556em); top: -5.201em; left: 50%; margin-left: -0.189em;"><span id="MathJax-Span-133" class="mtd"><span id="MathJax-Span-134" class="mrow"><span id="MathJax-Span-135" class="texatom"><span id="MathJax-Span-136" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-137" class="mi">j</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; clip: rect(2.434em, 1000em, 4.86em, -0.365em); top: -4.449em; left: 50%; margin-left: -0.853em;"><span id="MathJax-Span-152" class="mtd"><span id="MathJax-Span-153" class="mrow"><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-154" class="mfrac"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -3.37em; left: 50%; margin-left: -0.673em;"><span id="MathJax-Span-155" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-156" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-157" class="mi">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -2.004em; left: 50%; margin-left: -0.593em;"><span id="MathJax-Span-158" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-159" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-160" class="mi">u</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.467em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(2.434em, 1000em, 4.86em, -0.365em); top: -1.946em; left: 50%; margin-left: -0.853em;"><span id="MathJax-Span-173" class="mtd"><span id="MathJax-Span-174" class="mrow"><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-175" class="mfrac"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -3.37em; left: 50%; margin-left: -0.673em;"><span id="MathJax-Span-176" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-177" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-178" class="mi">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.182em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -2.004em; left: 50%; margin-left: -0.539em;"><span id="MathJax-Span-179" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-180" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-181" class="mi">v</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.467em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.897em;"></span></span><span style="position: absolute; clip: rect(2.887em, 1000em, 8.421em, -0.456em); top: -6.196em; left: 5.468em;"><span style="display: inline-block; position: relative; width: 0.593em; height: 0px;"><span style="position: absolute; clip: rect(1.892em, 1000em, 2.909em, -0.456em); top: -5.201em; left: 50%; margin-left: -0.296em;"><span id="MathJax-Span-138" class="mtd"><span id="MathJax-Span-139" class="mrow"><span id="MathJax-Span-140" class="texatom"><span id="MathJax-Span-141" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-142" class="mi">k</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; clip: rect(1.866em, 1000em, 2.878em, -0.446em); top: -3.156em; left: 50%; margin-left: -0.242em;"><span id="MathJax-Span-161" class="mtd"><span id="MathJax-Span-162" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-163" class="mn">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.866em, 1000em, 2.878em, -0.446em); top: -0.652em; left: 50%; margin-left: -0.242em;"><span id="MathJax-Span-182" class="mtd"><span id="MathJax-Span-183" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-184" class="mn">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.196em;"></span></span></span></span><span style="vertical-align: 3.253em;" id="MathJax-Span-185" class="mo"><span style="display: inline-block; position: relative; width: 0.278em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.237em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: 1.77em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -2.403em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -1.568em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -0.733em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.101em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: 0.936em; left: 0em;">∣<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 3.253em; vertical-align: 0em;"></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 3.253em; vertical-align: 0em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.711em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 7.092em; vertical-align: -3.256em;"></span></span></nobr></span></div><script id="MathJax-Element-3" type="math/tex; mode=display">\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
\end{vmatrix} </script>
<div class="math-header">The probability of getting <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-4-Frame" class="MathJax"><nobr><span id="MathJax-Span-186" class="math"><span style="display: inline-block; position: relative; width: 11px; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.614em, 1000em, 2.607em, -0.385em); top: -2.452em; left: 0em;"><span id="MathJax-Span-187" class="mrow"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-188" class="mi">k</span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.042em; vertical-align: -0.077em;"></span></span></nobr></span><script id="MathJax-Element-4" type="math/tex">k</script> heads when flipping <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-5-Frame" class="MathJax"><nobr><span id="MathJax-Span-189" class="math"><span style="display: inline-block; position: relative; width: 12px; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.607em, -0.412em); top: -2.452em; left: 0em;"><span id="MathJax-Span-190" class="mrow"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-191" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.714em; vertical-align: -0.077em;"></span></span></nobr></span><script id="MathJax-Element-5" type="math/tex">n</script> coins is</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-6-Frame" class="MathJax"><nobr><span id="MathJax-Span-192" class="math"><span style="display: inline-block; position: relative; width: 204px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(2.322em, 1000em, 5.044em, -0.452em); top: -3.933em; left: 0em;"><span id="MathJax-Span-193" class="mrow"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-194" class="mi">P<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.109em;"></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-195" class="mo">(</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-196" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.026em;"></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-197" class="mo">)</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-198" class="mo">=</span><span style="padding-left: 0.278em;" id="MathJax-Span-199" class="texatom"><span id="MathJax-Span-200" class="mrow"><span id="MathJax-Span-201" class="mfenced"><span style="vertical-align: 0em;" id="MathJax-Span-207" class="mo"><span style="font-family: MathJax_Size3;">(</span></span><span id="MathJax-Span-203" class="mfrac"><span style="display: inline-block; position: relative; width: 0.593em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.464em); top: -3.155em; left: 50%; margin-left: -0.296em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-204" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(1.623em, 1000em, 2.651em, -0.437em); top: -1.792em; left: 50%; margin-left: -0.269em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-205" class="mi">k</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span></span></span><span style="vertical-align: 0em;" id="MathJax-Span-208" class="mo"><span style="font-family: MathJax_Size3;">)</span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.45em; vertical-align: 0em;"></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.45em; vertical-align: 0em;"></span></span></span></span><span id="MathJax-Span-209" class="msubsup"><span style="display: inline-block; position: relative; width: 0.937em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.834em, -0.524em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-210" class="mi">p</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.73em; left: 0.485em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-211" class="mi">k</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-212" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-213" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-214" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-215" class="mi">p</span><span id="MathJax-Span-216" class="msubsup"><span style="display: inline-block; position: relative; width: 1.799em; height: 0px;"><span style="position: absolute; clip: rect(1.782em, 1000em, 3.106em, -0.43em); top: -2.694em; left: 0em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-217" class="mo">)</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; top: -2.848em; left: 0.377em;"><span id="MathJax-Span-218" class="texatom"><span id="MathJax-Span-219" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-220" class="mi">n</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-221" class="mo">−</span><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-222" class="mi">k</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.933em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 2.908em; vertical-align: -1.164em;"></span></span></nobr></span></div><script id="MathJax-Element-6" type="math/tex; mode=display">P(E) = {n \choose k} p^k (1-p)^{ n-k} </script>
<div class="math-header">An Identity of Ramanujan</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-7-Frame" class="MathJax"><nobr><span id="MathJax-Span-223" class="math"><span style="display: inline-block; position: relative; width: 312px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.022em, 1000em, 5.687em, -0.365em); top: -2.694em; left: 0em;"><span id="MathJax-Span-224" class="mrow"><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-225" class="mfrac"><span style="display: inline-block; position: relative; width: 7.609em; height: 0px;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.856em, -0.402em); top: -3.37em; left: 50%; margin-left: -0.242em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-226" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(2.46em, 1000em, 4.583em, -0.305em); top: -2.647em; left: 50%; margin-left: -3.745em;"><span id="MathJax-Span-227" class="mrow"><span id="MathJax-Span-228" class="texatom"><span id="MathJax-Span-229" class="mrow"><span style="vertical-align: 0em;" id="MathJax-Span-230" class="mo"><span style="font-family: MathJax_Size2;">(</span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.149em; vertical-align: 0em;"></span></span></span><span id="MathJax-Span-231" class="msqrt"><span style="display: inline-block; position: relative; width: 2.934em; height: 0px;"><span style="position: absolute; clip: rect(1.485em, 1000em, 2.845em, -0.442em); top: -2.478em; left: 1.024em;"><span id="MathJax-Span-232" class="mrow"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-233" class="mi">ϕ</span><span id="MathJax-Span-234" class="msqrt"><span style="display: inline-block; position: relative; width: 1.293em; height: 0px;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.878em, -0.435em); top: -2.694em; left: 0.808em;"><span id="MathJax-Span-235" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-236" class="mn">5</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.916em, 1000em, 1.307em, -0.485em); top: -1.909em; left: 0.808em;"><span style="border-left: 0.485em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: -0.067em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.025em, 1000em, 4.349em, -0.413em); top: -4.019em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(3.555em, 1000em, 4.019em, -0.485em); top: -4.64em; left: 1.024em;"><span style="display: inline-block; position: relative; width: 1.911em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.987em; left: -0.084em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.987em; left: 1.217em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.987em; left: 0.333em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.987em; left: 0.775em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(2.975em, 1000em, 4.499em, -0.374em); top: -4.06em; left: 0em;"><span style="font-family: MathJax_Size1;">√</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-237" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-238" class="mi">ϕ</span><span id="MathJax-Span-239" class="texatom"><span id="MathJax-Span-240" class="mrow"><span style="vertical-align: 0em;" id="MathJax-Span-241" class="mo"><span style="font-family: MathJax_Size2;">)</span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.149em; vertical-align: 0em;"></span></span></span><span id="MathJax-Span-242" class="msubsup"><span style="display: inline-block; position: relative; width: 1.584em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.446em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-243" class="mi">e</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.753em; left: 0.485em;"><span id="MathJax-Span-244" class="texatom"><span id="MathJax-Span-245" class="mrow"><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-246" class="mfrac"><span style="display: inline-block; position: relative; width: 0.389em; height: 0px;"><span style="position: absolute; clip: rect(1.768em, 1000em, 2.425em, -0.46em); top: -2.601em; left: 50%; margin-left: -0.135em;"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-247" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span><span style="position: absolute; clip: rect(1.768em, 1000em, 2.436em, -0.46em); top: -1.946em; left: 50%; margin-left: -0.135em;"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-248" class="mn">5</span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.224em; left: 0em;"><span style="border-left: 0.389em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-249" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.772em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 7.609em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-250" class="mo">=</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-251" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-252" class="mo">+</span><span style="padding-left: 0.342em; padding-right: 0.12em;" id="MathJax-Span-253" class="mfrac"><span style="display: inline-block; position: relative; width: 5.723em; height: 0px;"><span style="position: absolute; clip: rect(2.992em, 1000em, 4.16em, -0.446em); top: -4.664em; left: 50%; margin-left: -0.927em;"><span id="MathJax-Span-254" class="msubsup"><span style="display: inline-block; position: relative; width: 1.853em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.446em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-255" class="mi">e</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.734em; left: 0.485em;"><span id="MathJax-Span-256" class="texatom"><span id="MathJax-Span-257" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-258" class="mo">−</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-259" class="mn">2</span><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-260" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(1.571em, 1000em, 4.751em, -0.402em); top: -1.758em; left: 50%; margin-left: -2.802em;"><span id="MathJax-Span-261" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-262" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-263" class="mo">+</span><span style="padding-left: 0.342em; padding-right: 0.12em;" id="MathJax-Span-264" class="mfrac"><span style="display: inline-block; position: relative; width: 3.676em; height: 0px;"><span style="position: absolute; clip: rect(3.283em, 1000em, 4.156em, -0.457em); top: -4.406em; left: 50%; margin-left: -0.646em;"><span id="MathJax-Span-265" class="msubsup"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(1.843em, 1000em, 2.486em, -0.457em); top: -2.317em; left: 0em;"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-266" class="mi">e</span><span style="display: inline-block; width: 0px; height: 2.317em;"></span></span><span style="position: absolute; top: -2.467em; left: 0.323em;"><span id="MathJax-Span-267" class="texatom"><span id="MathJax-Span-268" class="mrow"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-269" class="mo">−</span><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-270" class="mn">4</span><span style="font-size: 50%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-271" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(1.388em, 1000em, 3.696em, -0.426em); top: -1.639em; left: 50%; margin-left: -1.778em;"><span id="MathJax-Span-272" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-273" class="mn">1</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-274" class="mo">+</span><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-275" class="mfrac"><span style="display: inline-block; position: relative; width: 2.383em; height: 0px;"><span style="position: absolute; clip: rect(3.348em, 1000em, 4.154em, -0.465em); top: -4.33em; left: 50%; margin-left: -0.584em;"><span id="MathJax-Span-276" class="msubsup"><span style="display: inline-block; position: relative; width: 1.169em; height: 0px;"><span style="position: absolute; clip: rect(1.88em, 1000em, 2.43em, -0.465em); top: -2.263em; left: 0em;"><span style="font-size: 50%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-277" class="mi">e</span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span><span style="position: absolute; top: -2.407em; left: 0.216em;"><span id="MathJax-Span-278" class="texatom"><span id="MathJax-Span-279" class="mrow"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-280" class="mo">−</span><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-281" class="mn">6</span><span style="font-size: 50%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-282" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(1.332em, 1000em, 2.835em, -0.443em); top: -1.51em; left: 50%; margin-left: -1.131em;"><span id="MathJax-Span-283" class="mrow"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-284" class="mn">1</span><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-285" class="mo">+</span><span style="padding-left: 0.12em; padding-right: 0.12em;" id="MathJax-Span-286" class="mfrac"><span style="display: inline-block; position: relative; width: 1.359em; height: 0px;"><span style="position: absolute; clip: rect(3.348em, 1000em, 4.154em, -0.465em); top: -4.279em; left: 50%; margin-left: -0.584em;"><span id="MathJax-Span-287" class="msubsup"><span style="display: inline-block; position: relative; width: 1.169em; height: 0px;"><span style="position: absolute; clip: rect(1.88em, 1000em, 2.43em, -0.465em); top: -2.263em; left: 0em;"><span style="font-size: 50%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-288" class="mi">e</span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span><span style="position: absolute; top: -2.407em; left: 0.216em;"><span id="MathJax-Span-289" class="texatom"><span id="MathJax-Span-290" class="mrow"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-291" class="mo">−</span><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-292" class="mn">8</span><span style="font-size: 50%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-293" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(1.768em, 1000em, 2.466em, -0.443em); top: -1.894em; left: 50%; margin-left: -0.62em;"><span id="MathJax-Span-294" class="mrow"><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-295" class="mn">1</span><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-296" class="mo">+</span><span style="font-size: 50%; font-family: MathJax_Main;" id="MathJax-Span-297" class="mo">…</span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.173em; left: 0em;"><span style="border-left: 1.359em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.224em; left: 0em;"><span style="border-left: 2.383em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 3.676em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 5.723em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 5.161em; vertical-align: -3.347em;"></span></span></nobr></span></div><script id="MathJax-Element-7" type="math/tex; mode=display"> \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } } </script>
<div class="math-header">A Rogers-Ramanujan Identity</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-8-Frame" class="MathJax"><nobr><span id="MathJax-Span-298" class="math"><span style="display: inline-block; position: relative; width: 640px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(0.97em, 1000em, 4.187em, -0.402em); top: -2.694em; left: 0em;"><span id="MathJax-Span-299" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-300" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-301" class="mo">+</span><span style="padding-left: 0.342em; padding-right: 0.12em;" id="MathJax-Span-302" class="mfrac"><span style="display: inline-block; position: relative; width: 3.029em; height: 0px;"><span style="position: absolute; clip: rect(2.992em, 1000em, 4.343em, -0.452em); top: -4.664em; left: 50%; margin-left: -0.469em;"><span id="MathJax-Span-303" class="msubsup"><span style="display: inline-block; position: relative; width: 0.937em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.834em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-304" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.734em; left: 0.485em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-305" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(1.782em, 1000em, 3.106em, -0.391em); top: -1.969em; left: 50%; margin-left: -1.455em;"><span id="MathJax-Span-306" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-307" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-308" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-309" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-310" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-311" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 3.029em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-312" class="mo">+</span><span style="padding-left: 0.342em; padding-right: 0.12em;" id="MathJax-Span-313" class="mfrac"><span style="display: inline-block; position: relative; width: 6.37em; height: 0px;"><span style="position: absolute; clip: rect(2.992em, 1000em, 4.343em, -0.452em); top: -4.664em; left: 50%; margin-left: -0.469em;"><span id="MathJax-Span-314" class="msubsup"><span style="display: inline-block; position: relative; width: 0.937em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.834em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-315" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.734em; left: 0.485em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-316" class="mn">6</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(1.773em, 1000em, 3.106em, -0.391em); top: -1.959em; left: 50%; margin-left: -3.125em;"><span id="MathJax-Span-317" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-318" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-319" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-320" class="mo">−</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-321" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-322" class="mo">)</span><span style="font-family: MathJax_Main;" id="MathJax-Span-323" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-324" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-325" class="mo">−</span><span style="padding-left: 0.222em;" id="MathJax-Span-326" class="msubsup"><span style="display: inline-block; position: relative; width: 0.937em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.834em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-327" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.66em; left: 0.485em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-328" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-329" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 6.37em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-330" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-331" class="mo">⋯</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-332" class="mo">=</span><span style="padding-left: 0.278em;" id="MathJax-Span-333" class="munderover"><span style="display: inline-block; position: relative; width: 1.293em; height: 0px;"><span style="position: absolute; clip: rect(2.66em, 1000em, 4.383em, -0.429em); top: -3.772em; left: 0em;"><span style="font-family: MathJax_Size2; vertical-align: 0em;" id="MathJax-Span-334" class="mo">∏</span><span style="display: inline-block; width: 0px; height: 3.772em;"></span></span><span style="position: absolute; clip: rect(1.738em, 1000em, 2.777em, -0.493em); top: -1.283em; left: 0.054em;"><span id="MathJax-Span-335" class="texatom"><span id="MathJax-Span-336" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-337" class="mi">j</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-338" class="mo">=</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-339" class="mn">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span><span style="position: absolute; clip: rect(1.797em, 1000em, 2.54em, -0.446em); top: -3.521em; left: 0.296em;"><span id="MathJax-Span-340" class="texatom"><span id="MathJax-Span-341" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-342" class="mi">∞</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="padding-left: 0.287em; padding-right: 0.12em;" id="MathJax-Span-343" class="mfrac"><span style="display: inline-block; position: relative; width: 9.226em; height: 0px;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.856em, -0.402em); top: -3.37em; left: 50%; margin-left: -0.242em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-344" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.773em, 1000em, 3.106em, -0.391em); top: -1.959em; left: 50%; margin-left: -4.553em;"><span id="MathJax-Span-345" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-346" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-347" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-348" class="mo">−</span><span style="padding-left: 0.222em;" id="MathJax-Span-349" class="msubsup"><span style="display: inline-block; position: relative; width: 2.123em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.834em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-350" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.66em; left: 0.485em;"><span id="MathJax-Span-351" class="texatom"><span id="MathJax-Span-352" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-353" class="mn">5</span><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-354" class="mi">j</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-355" class="mo">+</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-356" class="mn">2</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-357" class="mo">)</span><span style="font-family: MathJax_Main;" id="MathJax-Span-358" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-359" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-360" class="mo">−</span><span style="padding-left: 0.222em;" id="MathJax-Span-361" class="msubsup"><span style="display: inline-block; position: relative; width: 2.123em; height: 0px;"><span style="position: absolute; clip: rect(1.875em, 1000em, 2.834em, -0.452em); top: -2.478em; left: 0em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-362" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; top: -2.66em; left: 0.485em;"><span id="MathJax-Span-363" class="texatom"><span id="MathJax-Span-364" class="mrow"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-365" class="mn">5</span><span style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-366" class="mi">j</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-367" class="mo">+</span><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-368" class="mn">3</span></span></span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-369" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 9.226em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.167em;" id="MathJax-Span-370" class="mo">,</span><span style="height: 0em; vertical-align: 0em; width: 1em; display: inline-block; overflow: hidden;" id="MathJax-Span-371" class="mspace"></span><span style="height: 0em; vertical-align: 0em; width: 1em; display: inline-block; overflow: hidden;" id="MathJax-Span-372" class="mspace"></span><span style="font-family: MathJax_Main; padding-left: 0.167em;" id="MathJax-Span-373" class="mtext">for </span><span id="MathJax-Span-374" class="texatom"><span id="MathJax-Span-375" class="mrow"><span id="MathJax-Span-376" class="texatom"><span id="MathJax-Span-377" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-378" class="mo">|</span></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-379" class="mi">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.014em;"></span></span><span id="MathJax-Span-380" class="texatom"><span id="MathJax-Span-381" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-382" class="mo">|</span></span></span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-383" class="mo"><</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-384" class="mn">1</span></span></span><span style="font-family: MathJax_Main;" id="MathJax-Span-385" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 3.482em; vertical-align: -1.607em;"></span></span></nobr></span></div><script id="MathJax-Element-8" type="math/tex; mode=display"> 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},
\quad\quad \text{for $|q|<1$}. </script>
<div class="math-header">Maxwell’s Equations</div>
<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-9-Frame" class="MathJax"><nobr><span id="MathJax-Span-386" class="math"><span style="display: inline-block; position: relative; width: 202px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(-2.526em, 1000em, 6.552em, -0.272em); top: -2.263em; left: 0em;"><span id="MathJax-Span-387" class="mrow"><span style="padding-right: 0.167em; padding-left: 0.167em;" id="MathJax-Span-388" class="mtable"><span style="display: inline-block; position: relative; width: 10.569em; height: 0px;"><span style="position: absolute; clip: rect(4.801em, 1000em, 13.713em, -0.439em); top: -9.591em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.22em; height: 0px;"><span style="position: absolute; clip: rect(0.843em, 1000em, 3.567em, -0.439em); top: -5.632em; right: 0em;"><span id="MathJax-Span-389" class="mtd"><span id="MathJax-Span-390" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-391" class="mi">∇</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-392" class="mo">×</span><span style="padding-left: 0.222em;" id="MathJax-Span-393" class="texatom"><span id="MathJax-Span-394" class="mrow"><span id="MathJax-Span-395" class="munderover"><span style="display: inline-block; position: relative; width: 0.808em; height: 0px;"><span style="position: absolute; clip: rect(1.9em, 1000em, 2.909em, -0.446em); top: -2.748em; left: 0em;"><span id="MathJax-Span-396" class="texatom"><span id="MathJax-Span-397" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-398" class="mi">B</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.292em; left: 0.183em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-399" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-400" class="mo">−</span><span style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;" id="MathJax-Span-401" class="mspace"></span><span style="padding-left: 0.342em; padding-right: 0.12em;" id="MathJax-Span-402" class="mfrac"><span style="display: inline-block; position: relative; width: 0.605em; height: 0px;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.856em, -0.402em); top: -3.37em; left: 50%; margin-left: -0.242em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-403" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.451em); top: -1.792em; left: 50%; margin-left: -0.216em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-404" class="mi">c</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 0.605em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;" id="MathJax-Span-405" class="mspace"></span><span style="padding-left: 0.287em; padding-right: 0.12em;" id="MathJax-Span-406" class="mfrac"><span style="display: inline-block; position: relative; width: 1.467em; height: 0px;"><span style="position: absolute; clip: rect(1.52em, 1000em, 2.878em, -0.443em); top: -3.37em; left: 50%; margin-left: -0.673em;"><span id="MathJax-Span-407" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-408" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span id="MathJax-Span-409" class="texatom"><span id="MathJax-Span-410" class="mrow"><span id="MathJax-Span-411" class="munderover"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px;"><span style="position: absolute; clip: rect(1.906em, 1000em, 2.909em, -0.446em); top: -2.748em; left: 0em;"><span id="MathJax-Span-412" class="texatom"><span id="MathJax-Span-413" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-414" class="mi">E</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.286em; left: 0.156em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-415" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -2.004em; left: 50%; margin-left: -0.485em;"><span id="MathJax-Span-416" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-417" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-418" class="mi">t</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.467em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.52em, 1000em, 2.889em, -0.439em); top: -3.408em; right: 0em;"><span id="MathJax-Span-434" class="mtd"><span id="MathJax-Span-435" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-436" class="mi">∇</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-437" class="mo">⋅</span><span style="padding-left: 0.222em;" id="MathJax-Span-438" class="texatom"><span id="MathJax-Span-439" class="mrow"><span id="MathJax-Span-440" class="munderover"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px;"><span style="position: absolute; clip: rect(1.906em, 1000em, 2.909em, -0.446em); top: -2.748em; left: 0em;"><span id="MathJax-Span-441" class="texatom"><span id="MathJax-Span-442" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-443" class="mi">E</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.286em; left: 0.156em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-444" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.837em, 1000em, 3.567em, -0.439em); top: -0.997em; right: 0em;"><span id="MathJax-Span-451" class="mtd"><span id="MathJax-Span-452" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-453" class="mi">∇</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-454" class="mo">×</span><span style="padding-left: 0.222em;" id="MathJax-Span-455" class="texatom"><span id="MathJax-Span-456" class="mrow"><span id="MathJax-Span-457" class="munderover"><span style="display: inline-block; position: relative; width: 0.754em; height: 0px;"><span style="position: absolute; clip: rect(1.906em, 1000em, 2.909em, -0.446em); top: -2.748em; left: 0em;"><span id="MathJax-Span-458" class="texatom"><span id="MathJax-Span-459" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-460" class="mi">E</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.286em; left: 0.156em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-461" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span><span style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;" id="MathJax-Span-462" class="mspace"></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-463" class="mo">+</span><span style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;" id="MathJax-Span-464" class="mspace"></span><span style="padding-left: 0.342em; padding-right: 0.12em;" id="MathJax-Span-465" class="mfrac"><span style="display: inline-block; position: relative; width: 0.605em; height: 0px;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.856em, -0.402em); top: -3.37em; left: 50%; margin-left: -0.242em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-466" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.451em); top: -1.792em; left: 50%; margin-left: -0.216em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-467" class="mi">c</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 0.605em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="height: 0em; vertical-align: 0em; width: 0.167em; display: inline-block; overflow: hidden;" id="MathJax-Span-468" class="mspace"></span><span style="padding-left: 0.287em; padding-right: 0.12em;" id="MathJax-Span-469" class="mfrac"><span style="display: inline-block; position: relative; width: 1.521em; height: 0px;"><span style="position: absolute; clip: rect(1.514em, 1000em, 2.878em, -0.443em); top: -3.37em; left: 50%; margin-left: -0.7em;"><span id="MathJax-Span-470" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-471" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span id="MathJax-Span-472" class="texatom"><span id="MathJax-Span-473" class="mrow"><span id="MathJax-Span-474" class="munderover"><span style="display: inline-block; position: relative; width: 0.808em; height: 0px;"><span style="position: absolute; clip: rect(1.9em, 1000em, 2.909em, -0.446em); top: -2.748em; left: 0em;"><span id="MathJax-Span-475" class="texatom"><span id="MathJax-Span-476" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-477" class="mi">B</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.292em; left: 0.183em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-478" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.817em, 1000em, 2.878em, -0.443em); top: -2.004em; left: 50%; margin-left: -0.485em;"><span id="MathJax-Span-479" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-480" class="mi">∂<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.035em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-481" class="mi">t</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.521em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.514em, 1000em, 2.889em, -0.439em); top: 1.234em; right: 0em;"><span id="MathJax-Span-492" class="mtd"><span id="MathJax-Span-493" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-494" class="mi">∇</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-495" class="mo">⋅</span><span style="padding-left: 0.222em;" id="MathJax-Span-496" class="texatom"><span id="MathJax-Span-497" class="mrow"><span id="MathJax-Span-498" class="munderover"><span style="display: inline-block; position: relative; width: 0.808em; height: 0px;"><span style="position: absolute; clip: rect(1.9em, 1000em, 2.909em, -0.446em); top: -2.748em; left: 0em;"><span id="MathJax-Span-499" class="texatom"><span id="MathJax-Span-500" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-501" class="mi">B</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.292em; left: 0.183em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-502" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="display: inline-block; width: 0px; height: 9.591em;"></span></span><span style="position: absolute; clip: rect(4.814em, 1000em, 13.378em, -0.429em); top: -9.267em; left: 7.498em;"><span style="display: inline-block; position: relative; width: 3.071em; height: 0px;"><span style="position: absolute; clip: rect(1.179em, 1000em, 3.553em, -0.429em); top: -5.632em; left: 0em;"><span id="MathJax-Span-419" class="mtd"><span id="MathJax-Span-420" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-421" class="mo">=</span><span style="padding-left: 0.398em; padding-right: 0.12em;" id="MathJax-Span-422" class="mfrac"><span style="display: inline-block; position: relative; width: 1.198em; height: 0px;"><span style="position: absolute; clip: rect(1.855em, 1000em, 2.867em, -0.457em); top: -3.37em; left: 50%; margin-left: -0.539em;"><span id="MathJax-Span-423" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-424" class="mn">4</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-425" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.875em, 1000em, 2.651em, -0.451em); top: -1.792em; left: 50%; margin-left: -0.216em;"><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-426" class="mi">c</span><span style="display: inline-block; width: 0px; height: 2.478em;"></span></span><span style="position: absolute; clip: rect(0.849em, 1000em, 1.239em, -0.485em); top: -1.298em; left: 0em;"><span style="border-left: 1.198em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0em;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="padding-left: 0.167em;" id="MathJax-Span-427" class="texatom"><span id="MathJax-Span-428" class="mrow"><span id="MathJax-Span-429" class="munderover"><span style="display: inline-block; position: relative; width: 0.41em; height: 0px;"><span style="position: absolute; clip: rect(1.891em, 1000em, 3.109em, -0.556em); top: -2.748em; left: 0em;"><span id="MathJax-Span-430" class="texatom"><span id="MathJax-Span-431" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-432" class="mi">j</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.301em; left: -0.032em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-433" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.855em, 1000em, 3.072em, -0.429em); top: -3.408em; left: 0em;"><span id="MathJax-Span-445" class="mtd"><span id="MathJax-Span-446" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-447" class="mo">=</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-448" class="mn">4</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-449" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-450" class="mi">ρ</span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.546em, 1000em, 2.866em, -0.429em); top: -0.997em; left: 0em;"><span id="MathJax-Span-482" class="mtd"><span id="MathJax-Span-483" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-484" class="mo">=</span><span style="padding-left: 0.278em;" id="MathJax-Span-485" class="texatom"><span id="MathJax-Span-486" class="mrow"><span id="MathJax-Span-487" class="munderover"><span style="display: inline-block; position: relative; width: 0.593em; height: 0px;"><span style="position: absolute; clip: rect(1.932em, 1000em, 2.919em, -0.44em); top: -2.748em; left: 0em;"><span id="MathJax-Span-488" class="texatom"><span id="MathJax-Span-489" class="mrow"><span style="font-family: MathJax_Main; font-weight: bold;" id="MathJax-Span-490" class="mn">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.748em;"></span></span><span style="position: absolute; top: -4.26em; left: 0.075em;"><span style="height: 0em; vertical-align: 0em; width: 0.471em; display: inline-block; overflow: hidden;"></span><span style="font-family: MathJax_Main;" id="MathJax-Span-491" class="mo">⃗ <span style="height: 0em; vertical-align: 0em; margin-left: -0.269em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(1.866em, 1000em, 2.878em, -0.429em); top: 1.234em; left: 0em;"><span id="MathJax-Span-503" class="mtd"><span id="MathJax-Span-504" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-505" class="mo">=</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" id="MathJax-Span-506" class="mn">0</span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="display: inline-block; width: 0px; height: 9.267em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.263em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 10.281em; vertical-align: -4.851em;"></span></span></nobr></span></div><script id="MathJax-Element-9" type="math/tex; mode=display"> \begin{aligned}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
</script>
<p>Finally, while display equations look good for a page of samples, the
ability to mix math and text in a paragraph is also important. This
expression <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-10-Frame" class="MathJax"><nobr><span id="MathJax-Span-507" class="math"><span style="display: inline-block; position: relative; width: 153px; height: 0px; font-size: 116%;"><span style="position: absolute; clip: rect(1.584em, 1000em, 3.106em, -0.413em); top: -2.694em; left: 0em;"><span id="MathJax-Span-508" class="mrow"><span id="MathJax-Span-509" class="msqrt"><span style="display: inline-block; position: relative; width: 3.581em; height: 0px;"><span style="position: absolute; clip: rect(1.866em, 1000em, 2.878em, -0.443em); top: -2.694em; left: 0.808em;"><span id="MathJax-Span-510" class="mrow"><span style="font-family: MathJax_Main;" id="MathJax-Span-511" class="mn">3</span><span style="font-family: MathJax_Math; font-style: italic;" id="MathJax-Span-512" class="mi">x</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-513" class="mo">−</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-514" class="mn">1</span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; clip: rect(3.555em, 1000em, 4.019em, -0.485em); top: -4.549em; left: 0.808em;"><span style="display: inline-block; position: relative; width: 2.773em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.987em; left: -0.084em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.987em; left: 2.079em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.987em; left: 0.438em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.987em; left: 0.985em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.987em; left: 1.532em;">−<span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span><span style="position: absolute; clip: rect(3.025em, 1000em, 4.349em, -0.413em); top: -4.019em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 3.987em;"></span></span></span></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-515" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-516" class="mo">(</span><span style="font-family: MathJax_Main;" id="MathJax-Span-517" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" id="MathJax-Span-518" class="mo">+</span><span style="font-family: MathJax_Math; font-style: italic; padding-left: 0.222em;" id="MathJax-Span-519" class="mi">x</span><span id="MathJax-Span-520" class="msubsup"><span style="display: inline-block; position: relative; width: 0.829em; height: 0px;"><span style="position: absolute; clip: rect(1.782em, 1000em, 3.106em, -0.43em); top: -2.694em; left: 0em;"><span style="font-family: MathJax_Main;" id="MathJax-Span-521" class="mo">)</span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span><span style="position: absolute; top: -2.848em; left: 0.377em;"><span style="font-size: 70.7%; font-family: MathJax_Main;" id="MathJax-Span-522" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.371em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.694em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.515em; vertical-align: -0.352em;"></span></span></nobr></span><script id="MathJax-Element-10" type="math/tex">\sqrt{3x-1}+(1+x)^2</script>
is an example of an inline equation. As you see, MathJax equations can
be used this way as well, without unduly disturbing the spacing between
lines.</p> </div><!-- .entry-content -->
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<a target="_blank" href="http://twitter.com/intent/user?screen_name=MathJax" class="twtr-user">MathJax</a> @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=mollydotcom">mollydotcom</a> you could probably do this via MathJax and the third party extension called xyjax <a class="twtr-hyperlink" target="_blank" href="http://t.co/KKswoc6I">http://t.co/KKswoc6I</a>
<em>
<a target="_blank" class="twtr-timestamp" time="Sat, 22 Dec 2012 00:00:57 +0000" href="http://twitter.com/MathJax/status/282274488215822339">yesterday</a> ·
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<a target="_blank" href="http://twitter.com/intent/user?screen_name=pkrautz" class="twtr-user">pkrautz</a> @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=Klortho">Klortho</a> thank you very much for your bug tracking on the @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=MathJax">MathJax</a> user group! This <a class="twtr-hyperlink" target="_blank" href="http://t.co/ekU8KiAX">http://t.co/ekU8KiAX</a> helped a lot!
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<a target="_blank" class="twtr-timestamp" time="Sat, 22 Dec 2012 00:55:54 +0000" href="http://twitter.com/pkrautz/status/282288319759523840">yesterday</a> ·
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<a target="_blank" href="http://twitter.com/intent/user?screen_name=cartazio" class="twtr-user">cartazio</a> @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=smdiehl">smdiehl</a> they use mathjax !
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<a target="_blank" class="twtr-timestamp" time="Sat, 22 Dec 2012 08:00:35 +0000" href="http://twitter.com/cartazio/status/282395192542253056">yesterday</a> ·
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<a target="_blank" href="http://twitter.com/intent/user?screen_name=isinf" class="twtr-user">isinf</a> Woohoo! I've just added <a target="_blank" class="twtr-hashtag" href="http://twitter.com/search?q=%23MathJax">#MathJax</a> support to my markdown blog :)
<em>
<a target="_blank" class="twtr-timestamp" time="Sat, 22 Dec 2012 17:25:37 +0000" href="http://twitter.com/isinf/status/282537388478177280">20 hours ago</a> ·
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<a target="_blank" href="http://twitter.com/intent/user?screen_name=iSpencerX" class="twtr-user">iSpencerX</a> @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=MathJax">MathJax</a> what would that look like when modelled as a solid object? @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=sgi_corp">sgi_corp</a> @<a target="_blank" class="twtr-atreply" href="http://twitter.com/intent/user?screen_name=mollydotcom">mollydotcom</a> <a target="_blank" class="twtr-hashtag" href="http://twitter.com/search?q=%23math">#math</a> <a target="_blank" class="twtr-hashtag" href="http://twitter.com/search?q=%23hpc">#hpc</a>(cloud)
<em>
<a target="_blank" class="twtr-timestamp" time="Sun, 23 Dec 2012 14:00:04 +0000" href="http://twitter.com/iSpencerX/status/282848050085523456">22 minutes ago</a> ·
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<a href="http://www.mathjax.org/2012/12/17/news/the-london-mathematical-society-continues-as-mathjax-supporter/" title="The London Mathematical Society continues as MathJax supporter">The London Mathematical Society continues as MathJax supporter</a>
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<a href="http://www.mathjax.org/2012/11/14/news/iop-publishing-continues-as-mathjax-supporter/" title="IOP Publishing continues as MathJax Supporter">IOP Publishing continues as MathJax Supporter</a>
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<a href="http://www.mathjax.org/2012/11/09/news/mathjax-community-meetup-at-the-joint-mathematics-meetings-2013/" title="MathJax community meetup at the Joint Mathematics Meetings 2013">MathJax community meetup at the Joint Mathematics Meetings 2013</a>
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<a href="http://www.mathjax.org/2012/11/01/news/project-euclid-continues-as-mathjax-supporter/" title="Project Euclid continues as MathJax Supporter">Project Euclid continues as MathJax Supporter</a>
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<a href="http://www.mathjax.org/2012/10/22/news/mathjax-v2-1-now-available/" title="MathJax v2.1 now available">MathJax v2.1 now available</a>
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