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<title>Generating function (physics)</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generating function (physics)</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">This article is about generating functions in physics. For generating functions in mathematics, see <a href="Generating_function" title="Generating function">Generating function</a>.</div>
<p>In physics, and more specifically in <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, a <b>generating function</b> is, loosely, a function whose partial derivatives generate the differential equations that determine a system's dynamics. Common examples are the <a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a> of <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a <a href="Canonical_transformation" title="Canonical transformation">canonical transformation</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="In_canonical_transformations">In canonical transformations</h2></div>
<p>There are four basic generating functions, summarized by the following table:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<table class="wikitable" style="margin-left:1.5em;">
<tbody><tr>
<th style="background:#ffdead;">Generating function
</th>
<th style="background:#ffdead;">Its derivatives
</th></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=F_{1}(q,Q,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>=</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo>,</mo>
<mi>Q</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F=F_{1}(q,Q,t)}</annotation>
</semantics>
</math></span><img src="./011d2eed1670abc0e08573555306930dfcedcf74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.013ex; height:2.843ex;" alt="{\displaystyle F=F_{1}(q,Q,t)}" loading="lazy"></span>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=~~{\frac {\partial F_{1}}{\partial q}}\,\!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mtext>&nbsp;</mtext>
<mtext>&nbsp;</mtext>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>q</mi>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=~~{\frac {\partial F_{1}}{\partial q}}\,\!}</annotation>
</semantics>
</math></span><img src="./deaecd12a10db862a25ed8823d1c2918b7a88194.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; margin-right: -0.387ex; width:10.609ex; height:5.843ex;" alt="{\displaystyle p=~~{\frac {\partial F_{1}}{\partial q}}\,\!}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=-{\frac {\partial F_{1}}{\partial Q}}\,\!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Q</mi>
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<mspace width="thinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle P=-{\frac {\partial F_{1}}{\partial Q}}\,\!}</annotation>
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</math></span><img src="./9bd334b53cdcb9d25ed16bd6685d36c3d7e0dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.387ex; width:11.742ex; height:5.843ex;" alt="{\displaystyle P=-{\frac {\partial F_{1}}{\partial Q}}\,\!}" loading="lazy"></span>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F&amp;=F_{2}(q,P,t)\\&amp;=F_{1}+QP\end{aligned}}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mtr>
<mtd>
<mi>F</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>F</mi>
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<mn>2</mn>
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<mo stretchy="false">)</mo>
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<mtd>
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<mi>P</mi>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F&amp;=F_{2}(q,P,t)\\&amp;=F_{1}+QP\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./1237f4831eacdeec93115af598fb2a363aa7d68e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.671ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}F&amp;=F_{2}(q,P,t)\\&amp;=F_{1}+QP\end{aligned}}}" loading="lazy"></span>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=~~{\frac {\partial F_{2}}{\partial q}}\,\!}">
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<mfrac>
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<mn>2</mn>
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<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>q</mi>
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<mspace width="thinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle p=~~{\frac {\partial F_{2}}{\partial q}}\,\!}</annotation>
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</math></span><img src="./f3606411815a8c608e8b85f1158af15b4d856831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; margin-right: -0.387ex; width:10.609ex; height:5.843ex;" alt="{\displaystyle p=~~{\frac {\partial F_{2}}{\partial q}}\,\!}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=~~{\frac {\partial F_{2}}{\partial P}}\,\!}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mi>F</mi>
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<mn>2</mn>
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<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<mspace width="thinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle Q=~~{\frac {\partial F_{2}}{\partial P}}\,\!}</annotation>
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</math></span><img src="./3ef44186b4dc5815dabc8ca0c861a8aa503816fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:11.188ex; height:5.509ex;" alt="{\displaystyle Q=~~{\frac {\partial F_{2}}{\partial P}}\,\!}" loading="lazy"></span>
</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F&amp;=F_{3}(p,Q,t)\\&amp;=F_{1}-qp\end{aligned}}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mtr>
<mtd>
<mi>F</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msub>
<mo stretchy="false">(</mo>
<mi>p</mi>
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<mo>,</mo>
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<mo stretchy="false">)</mo>
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<mtr>
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<mtd>
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<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>−<!-- − --></mo>
<mi>q</mi>
<mi>p</mi>
</mtd>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F&amp;=F_{3}(p,Q,t)\\&amp;=F_{1}-qp\end{aligned}}}</annotation>
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</math></span><img src="./e3896e0eaaf89c50d5d523d7708a769056dae2e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.864ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}F&amp;=F_{3}(p,Q,t)\\&amp;=F_{1}-qp\end{aligned}}}" loading="lazy"></span>
</td>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}\,\!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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<mrow>
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<annotation encoding="application/x-tex">{\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}\,\!}</annotation>
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo>=</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Q</mi>
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<mspace width="thinmathspace"></mspace>
<mspace width="negativethinmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}}\,\!}</annotation>
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</td></tr>
<tr>
<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F&amp;=F_{4}(p,P,t)\\&amp;=F_{1}-qp+QP\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F&amp;=F_{4}(p,P,t)\\&amp;=F_{1}-qp+QP\end{aligned}}}</annotation>
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<td><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=-{\frac {\partial F_{4}}{\partial p}}\,\!}">
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<annotation encoding="application/x-tex">{\displaystyle q=-{\frac {\partial F_{4}}{\partial p}}\,\!}</annotation>
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<msub>
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<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<mi>P</mi>
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<annotation encoding="application/x-tex">{\displaystyle Q=~~{\frac {\partial F_{4}}{\partial P}}\,\!}</annotation>
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</math></span><img src="./961f7453d2f66ca8c0f1e20a8b7ddf8d13cea994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:11.188ex; height:5.509ex;" alt="{\displaystyle Q=~~{\frac {\partial F_{4}}{\partial P}}\,\!}" loading="lazy"></span>
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>
<p>Sometimes a given Hamiltonian can be turned into one that looks like the <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> Hamiltonian, which is
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=aP^{2}+bQ^{2}.}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
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<annotation encoding="application/x-tex">{\displaystyle H=aP^{2}+bQ^{2}.}</annotation>
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</math></span></span>
</p><p>For example, with the Hamiltonian
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {1}{2q^{2}}}+{\frac {p^{2}q^{4}}{2}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
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<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{2q^{2}}}+{\frac {p^{2}q^{4}}{2}},}</annotation>
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</math></span></span>
</p><p>where <span class="texhtml mvar" style="font-style:italic;">p</span> is the generalized momentum and <span class="texhtml mvar" style="font-style:italic;">q</span> is the <a href="Generalized_coordinates" title="Generalized coordinates">generalized coordinate</a>, a good canonical transformation to choose would be
</p>
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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=pq^{2}{\text{ and }}Q={\frac {-1}{q}}.}">
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<annotation encoding="application/x-tex">{\displaystyle P=pq^{2}{\text{ and }}Q={\frac {-1}{q}}.}</annotation>
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</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>
<p>This turns the Hamiltonian into
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {Q^{2}}{2}}+{\frac {P^{2}}{2}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
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<mfrac>
<msup>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle H={\frac {Q^{2}}{2}}+{\frac {P^{2}}{2}},}</annotation>
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</math></span></span>
</p><p>which is in the form of the harmonic oscillator Hamiltonian.
</p><p>The generating function <span class="texhtml"><i>F</i></span> for this transformation is of the third kind,
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=F_{3}(p,Q).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>F</mi>
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<annotation encoding="application/x-tex">{\displaystyle F=F_{3}(p,Q).}</annotation>
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</math></span></span>
</p><p>To find <span class="texhtml"><i>F</i></span> explicitly, use the equation for its derivative from the table above,
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}},}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}},}</annotation>
</semantics>
</math></span></span>
</p><p>and substitute the expression for <span class="texhtml mvar" style="font-style:italic;">P</span> from equation (<b><a href="#math_1">1</a></b>), expressed in terms of <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">Q</span>:
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p}{Q^{2}}}=-{\frac {\partial F_{3}}{\partial Q}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {p}{Q^{2}}}=-{\frac {\partial F_{3}}{\partial Q}}}</annotation>
</semantics>
</math></span></span>
</p><p>Integrating this with respect to <span class="texhtml mvar" style="font-style:italic;">Q</span> results in an equation for the generating function of the transformation given by equation (<b><a href="#math_1">1</a></b>):
</p>
<div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: var(--color-success,#14866d); color: inherit;text-align: center; display: table">
<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}(p,Q)={\frac {p}{Q}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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</msub>
<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle F_{3}(p,Q)={\frac {p}{Q}}}</annotation>
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</math></span><img src="./2a2cccdca0944a83f2d84e214cde92911d237cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.173ex; height:5.343ex;" alt="{\displaystyle F_{3}(p,Q)={\frac {p}{Q}}}" loading="lazy"></span>
</p>
</div>
<p>To confirm that this is the correct generating function, verify that it matches (<b><a href="#math_1">1</a></b>):
</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}={\frac {-1}{Q}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}={\frac {-1}{Q}}}</annotation>
</semantics>
</math></span></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li>
<li><a href="Poisson_bracket" title="Poisson bracket">Poisson bracket</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFGoldsteinPooleSafko2001" class="citation book cs1">Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). <i>Classical Mechanics</i> (3rd&nbsp;ed.). Addison-Wesley. p.&nbsp;373. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-0-201-65702-9</bdi>.</cite></span>
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