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Dilogarithm
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<h2 id="analytic-structure">Analytic structure</h2> <p>Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at z = 1 {\displaystyle z=1} , where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis ( 1 , ∞ ) {\displaystyle (1,\infty )} . However, the function is continuous at the branch point and takes on the value Li 2 ( 1 ) = π 2 / 6 {\displaystyle \operatorname {Li} _{2}(1)=\pi ^{2}/6} . </p> <h2 id="identities">Identities</h2> Li 2 ( z ) + Li 2 ( − z ) = 1 2 Li 2 ( z 2 ) . {\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2}).} <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Li 2 ( 1 − z ) + Li 2 ( 1 − 1 z ) = − ( ln z ) 2 2 . {\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {(\ln z)^{2}}{2}}.} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Li 2 ( z ) + Li 2 ( 1 − z ) = π 2 6 − ln z ⋅ ln ( 1 − z ) . {\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z).} <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> The <a href="/facts/Reflection_formula/aKUhTbH0">reflection formula</a>. Li 2 ( − z ) − Li 2 ( 1 − z ) + 1 2 Li 2 ( 1 − z 2 ) = − π 2 12 − ln z ⋅ ln ( z + 1 ) . {\displaystyle \operatorname {Li} _{2}(-z)-\operatorname {Li} _{2}(1-z)+{\frac {1}{2}}\operatorname {Li} _{2}(1-z^{2})=-{\frac {{\pi }^{2}}{12}}-\ln z\cdot \ln(z+1).} <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Li 2 ( z ) + Li 2 ( 1 z ) = − π 2 6 − ( ln ( − z ) ) 2 2 . {\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}\left({\frac {1}{z}}\right)=-{\frac {\pi ^{2}}{6}}-{\frac {(\ln(-z))^{2}}{2}}.} <a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> L ( x ) + L ( y ) = L ( x y ) + L ( x ( 1 − y ) 1 − x y ) + L ( y ( 1 − x ) 1 − x y ) {\displaystyle \operatorname {L} (x)+\operatorname {L} (y)=\operatorname {L} (xy)+\operatorname {L} ({\frac {x(1-y)}{1-xy}})+\operatorname {L} ({\frac {y(1-x)}{1-xy}})} .<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> Abel's functional equation or five-term relation where L ( z ) = π 6 [ Li 2 ( z ) + 1 2 ln ( z ) ln ( 1 − z ) ] {\displaystyle \operatorname {L} (z)={\frac {\pi }{6}}[\operatorname {Li} _{2}(z)+{\frac {1}{2}}\ln(z)\ln(1-z)]} is the <a href="/facts/Leonard_James_Rogers/ldnvylwH">Rogers</a> L-function (an analogous relation is satisfied also by the <a href="/facts/Quantum_dilogarithm/wLgEG1ct">quantum dilogarithm</a>) <h2 id="particular-value-identities">Particular value identities</h2> Li 2 ( 1 3 ) − 1 6 Li 2 ( 1 9 ) = π 2 18 − ( ln 3 ) 2 6 . {\displaystyle \operatorname {Li} _{2}\left({\frac {1}{3}}\right)-{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}-{\frac {(\ln 3)^{2}}{6}}.} <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> Li 2 ( − 1 3 ) − 1 3 Li 2 ( 1 9 ) = − π 2 18 + ( ln 3 ) 2 6 . {\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{3}}\right)-{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+{\frac {(\ln 3)^{2}}{6}}.} <a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Li 2 ( − 1 2 ) + 1 6 Li 2 ( 1 9 ) = − π 2 18 + ln 2 ⋅ ln 3 − ( ln 2 ) 2 2 − ( ln 3 ) 2 3 . {\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{2}}\right)+{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+\ln 2\cdot \ln 3-{\frac {(\ln 2)^{2}}{2}}-{\frac {(\ln 3)^{2}}{3}}.} <a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Li 2 ( 1 4 ) + 1 3 Li 2 ( 1 9 ) = π 2 18 + 2 ln 2 ⋅ ln 3 − 2 ( ln 2 ) 2 − 2 3 ( ln 3 ) 2 . {\displaystyle \operatorname {Li} _{2}\left({\frac {1}{4}}\right)+{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}+2\ln 2\cdot \ln 3-2(\ln 2)^{2}-{\frac {2}{3}}(\ln 3)^{2}.} <a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> Li 2 ( − 1 8 ) + Li 2 ( 1 9 ) = − 1 2 ( ln 9 8 ) 2 . {\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{8}}\right)+\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {1}{2}}\left(\ln {\frac {9}{8}}\right)^{2}.} <a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> 36 Li 2 ( 1 2 ) − 36 Li 2 ( 1 4 ) − 12 Li 2 ( 1 8 ) + 6 Li 2 ( 1 64 ) = π 2 . {\displaystyle 36\operatorname {Li} _{2}\left({\frac {1}{2}}\right)-36\operatorname {Li} _{2}\left({\frac {1}{4}}\right)-12\operatorname {Li} _{2}\left({\frac {1}{8}}\right)+6\operatorname {Li} _{2}\left({\frac {1}{64}}\right)={\pi }^{2}.} <h2 id="special-values">Special values</h2> Li 2 ( − 1 ) = − π 2 12 . {\displaystyle \operatorname {Li} _{2}(-1)=-{\frac {{\pi }^{2}}{12}}.} Li 2 ( 0 ) = 0. {\displaystyle \operatorname {Li} _{2}(0)=0.} Its slope = 1. Li 2 ( 1 2 ) = π 2 12 − ( ln 2 ) 2 2 . {\displaystyle \operatorname {Li} _{2}\left({\frac {1}{2}}\right)={\frac {{\pi }^{2}}{12}}-{\frac {(\ln 2)^{2}}{2}}.} Li 2 ( 1 ) = ζ ( 2 ) = π 2 6 , {\displaystyle \operatorname {Li} _{2}(1)=\zeta (2)={\frac {{\pi }^{2}}{6}},} where ζ ( s ) {\displaystyle \zeta (s)} is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>. Li 2 ( 2 ) = π 2 4 − i π ln 2. {\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2.} Li 2 ( − 5 − 1 2 ) = − π 2 15 + 1 2 ( ln 5 + 1 2 ) 2 = − π 2 15 + 1 2 arcsch 2 2. {\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)&=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\left(\ln {\frac {{\sqrt {5}}+1}{2}}\right)^{2}\\&=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\operatorname {arcsch} ^{2}2.\end{aligned}}} Li 2 ( − 5 + 1 2 ) = − π 2 10 − ln 2 5 + 1 2 = − π 2 10 − arcsch 2 2. {\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}+1}{2}}\right)&=-{\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&=-{\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2.\end{aligned}}} Li 2 ( 3 − 5 2 ) = π 2 15 − ln 2 5 + 1 2 = π 2 15 − arcsch 2 2. {\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {3-{\sqrt {5}}}{2}}\right)&={\frac {{\pi }^{2}}{15}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&={\frac {{\pi }^{2}}{15}}-\operatorname {arcsch} ^{2}2.\end{aligned}}} Li 2 ( 5 − 1 2 ) = π 2 10 − ln 2 5 + 1 2 = π 2 10 − arcsch 2 2. {\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {{\sqrt {5}}-1}{2}}\right)&={\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}\\&={\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2.\end{aligned}}} <h2 id="in-particle-physics">In particle physics</h2> <p>Spence's Function is commonly encountered in particle physics while calculating <a href="/facts/Radiative_correction/B4PdTB4o">radiative corrections</a>. In this context, the function is often defined with an absolute value inside the logarithm: </p> Φ ( x ) = − ∫ 0 x ln | 1 − u | u d u = { Li 2 ( x ) , x ≤ 1 ; π 2 3 − 1 2 ( ln x ) 2 − Li 2 ( 1 x ) , x > 1. {\displaystyle \operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u}}\,du={\begin{cases}\operatorname {Li} _{2}(x),&x\leq 1;\\{\frac {\pi ^{2}}{3}}-{\frac {1}{2}}(\ln x)^{2}-\operatorname {Li} _{2}({\frac {1}{x}}),&x>1.\end{cases}}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Markstein_number/YfsIPnt2">Markstein number</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Lewin, L. (1958). <i>Dilogarithms and associated functions</i>. Foreword by J. C. P. Miller. London: Macdonald. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0105524">0105524</a>.</li> <li>Morris, Robert (1979). <a href="https://doi.org/10.1090%2FS0025-5718-1979-0521291-X">"The dilogarithm function of a real argument"</a>. <i>Math. Comp</i>. 33 (146): 778–787. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0025-5718-1979-0521291-X">10.1090/S0025-5718-1979-0521291-X</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0521291">0521291</a>.</li> <li>Loxton, J. H. (1984). <a href="http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-aav43i2p155bwm?q=bwmeta1.element.bwnjournal-number-aa-1983-1984-43-2&qt=CHILDREN-STATELESS">"Special values of the dilogarithm"</a>. <i>Acta Arith</i>. 18 (2): 155–166. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4064%2Faa-43-2-155-166">10.4064/aa-43-2-155-166</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0736728">0736728</a>.</li> <li>Kirillov, Anatol N. (1995). "Dilogarithm identities". <i>Progress of Theoretical Physics Supplement</i>. 118: 61–142. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9408113">hep-th/9408113</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995PThPS.118...61K">1995PThPS.118...61K</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1143%2FPTPS.118.61">10.1143/PTPS.118.61</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119177149">119177149</a>.</li> <li>Osacar, Carlos; Palacian, Jesus; Palacios, Manuel (1995). "Numerical evaluation of the dilogarithm of complex argument". <i>Celest. Mech. Dyn. Astron</i>. 62 (1): 93–98. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995CeMDA..62...93O">1995CeMDA..62...93O</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00692071">10.1007/BF00692071</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121304484">121304484</a>.</li> <li>Zagier, Don (2007). "The Dilogarithm Function". In Pierre Cartier; Pierre Moussa; Bernard Julia; Pierre Vanhove (eds.). <a href="http://maths.dur.ac.uk/~dma0hg/dilog.pdf"><i>Frontiers in Number Theory, Physics, and Geometry II</i></a> (PDF). pp. 3–65. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-30308-4_1">10.1007/978-3-540-30308-4_1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-30308-4.</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Spencer_Bloch/xywM6bIw">Bloch, Spencer J.</a> (2000). <i>Higher regulators, algebraic </i>K<i>-theory, and zeta functions of elliptic curves</i>. CRM Monograph Series. Vol. 11. Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-2114-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0958.19001">0958.19001</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://dlmf.nist.gov/25.12">NIST Digital Library of Mathematical Functions: Dilogarithm</a></li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Dilogarithm.html">"Dilogarithm"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Zagier p. 10 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"William Spence - Biography". <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Spence/" target="_blank">https://mathshistory.st-andrews.ac.uk/Biographies/Spence/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography". <a href="http://www.biographi.ca/009004-119.01-e.php?BioId=37522" target="_blank">http://www.biographi.ca/009004-119.01-e.php?BioId=37522</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Zagier <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Zagier <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Zagier <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Weisstein, Eric W. "Rogers L-Function". mathworld.wolfram.com. Retrieved 2024-08-01. <a href="https://mathworld.wolfram.com/" target="_blank">https://mathworld.wolfram.com/</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Rogers, L. J. (1907). "On the Representation of Certain Asymptotic Series as Convergent Continued Fractions". Proceedings of the London Mathematical Society. s2-4 (1): 72–89. doi:10.1112/plms/s2-4.1.72. <a href="http://doi.wiley.com/10.1112/plms/s2-4.1.72" target="_blank">http://doi.wiley.com/10.1112/plms/s2-4.1.72</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Weisstein, Eric W. "Dilogarithm". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>