Spence-függvény

Matematikában, a Spence-függvény, vagy dilogaritmus, egy speciális függvény, mely a polilogaritmus egy speciális esete. Jelölése: Li₂(z). A Lobacsevszkij-függvény, és a Clausen-függvény szorosan kapcsolódik a Spence-függvényhez. E két függvényt is, és magát a dilogaritmust is nevezik Spence-függvénynek:

$\operatorname {Li} _{2}(\pm z)=-\int _{0}^{z}{\ln |1\mp \zeta | \over \zeta }\,\mathrm {d} \zeta =\sum _{k=1}^{\infty }{(\pm z)^{k} \over k^{2}};$

A függvényt William Spence (1777 – 1815), skót matematikusról nevezték el.

Kapcsolódó azonosságok

$\operatorname {Li} _{2}(-z)=-\operatorname {Li} _{2}\left({\frac {z}{1+z}}\right)-{\frac {\ln ^{2}(1+z)}{2}}$

$\operatorname {Li} _{2}({\rm {i}}z)={\frac {\operatorname {Li} _{2}(-z^{2})}{4}}+{\rm {i}}\operatorname {Li} _{2}(z)$

$\operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2})$

$\operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {\ln ^{2}z}{2}}$

$\operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z)$

$\operatorname {Li} _{2}(-z)-\operatorname {Li} _{2}(1-z)+{\frac {1}{2}}\operatorname {Li} _{2}(1-z^{2})=-{\frac {{\pi }^{2}}{12}}-\ln z$

$\operatorname {Li} _{2}\left({\frac {1}{3}}\right)-{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}-\ln ^{2}3$

$\operatorname {Li} _{2}\left(-{\frac {1}{2}}\right)+{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}-\ln 2\cdot \ln 3-{\frac {\ln ^{2}2}{2}}-{\frac {\ln ^{2}3}{3}}$

$\operatorname {Li} _{2}\left({\frac {1}{4}}\right)+{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}+2\ln 2\ln 3-2\ln ^{2}2-{\frac {2}{3}}\ln ^{2}3$

$\operatorname {Li} _{2}\left(-{\frac {1}{3}}\right)-{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+{\frac {1}{6}}\ln ^{2}3$

$\operatorname {Li} _{2}\left(-{\frac {1}{8}}\right)+\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {1}{2}}\ln ^{2}{\frac {9}{8}}$

$36\operatorname {Li} _{2}\left({\frac {1}{2}}\right)-36\operatorname {Li} _{2}\left({\frac {1}{4}}\right)-12\operatorname {Li} _{2}\left({\frac {1}{8}}\right)+6\operatorname {Li} _{2}\left({\frac {1}{64}}\right)={\pi }^{2}$

Speciális értékek

$\operatorname {Li} _{2}(-1)=-{\frac {{\pi }^{2}}{12}}$

$\operatorname {Li} _{2}(0)=0$

$\operatorname {Li} _{2}\left({\frac {1}{2}}\right)={\frac {{\pi }^{2}}{12}}-{\frac {\ln ^{2}2}{2}}$

$\operatorname {Li} _{2}(1)={\frac {{\pi }^{2}}{6}}$

$\operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}$

$\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)=-{\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}$

$=-{\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2$

$\operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}+1}{2}}\right)=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}$

$=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\operatorname {arcsch} ^{2}2$

$\operatorname {Li} _{2}\left({\frac {3+{\sqrt {5}}}{2}}\right)={\frac {{\pi }^{2}}{15}}-{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}$

$={\frac {{\pi }^{2}}{15}}-{\frac {1}{2}}\operatorname {arcsch} ^{2}2$

$\operatorname {Li} _{2}\left({\frac {{\sqrt {5}}+1}{2}}\right)={\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}$

$={\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2$

Források

• Lewin, L: Dilogarithms and associated functions. 1958.  
• Morris, Robert: "The dilogarithm function of a real argument". (hely nélkül): Math. Comp. 33. 1979. 778–787. o.  
• Kirillov, Anatol N: Dilogarithm identities. 1994.  

Kapcsolódó szócikkek

• http://maths.dur.ac.uk/~dma0hg/dilog.pdf
• http://mathworld.wolfram.com/Dilogarithm.html
• Lobacsevszkij-függvény
• Clausen-függvény
• Polilogaritmus