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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:
Li
2
(
±
z
)
=
−
∫
0
z
ln
|
1
∓
ζ
|
ζ
d
ζ
=
∑
k
=
1
∞
(
±
z
)
k
k
2
;
{\displaystyle \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.[ 1] [ 2]
Li
2
(
−
z
)
=
−
Li
2
(
z
1
+
z
)
−
ln
2
(
1
+
z
)
2
{\displaystyle \operatorname {Li} _{2}(-z)=-\operatorname {Li} _{2}\left({\frac {z}{1+z}}\right)-{\frac {\ln ^{2}(1+z)}{2}}}
Li
2
(
i
z
)
=
Li
2
(
−
z
2
)
4
+
i
Li
2
(
z
)
{\displaystyle \operatorname {Li} _{2}({\rm {i}}z)={\frac {\operatorname {Li} _{2}(-z^{2})}{4}}+{\rm {i}}\operatorname {Li} _{2}(z)}
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})}
Li
2
(
1
−
z
)
+
Li
2
(
1
−
1
z
)
=
−
ln
2
z
2
{\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {\ln ^{2}z}{2}}}
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)}
Li
2
(
−
z
)
−
Li
2
(
1
−
z
)
+
1
2
Li
2
(
1
−
z
2
)
=
−
π
2
12
−
ln
z
{\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}
Li
2
(
1
3
)
−
1
6
Li
2
(
1
9
)
=
π
2
18
−
ln
2
3
{\displaystyle \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}
Li
2
(
−
1
2
)
+
1
6
Li
2
(
1
9
)
=
−
π
2
18
−
ln
2
⋅
ln
3
−
ln
2
2
2
−
ln
2
3
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 ^{2}3}{3}}}
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
2
3
{\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\ln 3-2\ln ^{2}2-{\frac {2}{3}}\ln ^{2}3}
Li
2
(
−
1
3
)
−
1
3
Li
2
(
1
9
)
=
−
π
2
18
+
1
6
ln
2
3
{\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 {1}{6}}\ln ^{2}3}
Li
2
(
−
1
8
)
+
Li
2
(
1
9
)
=
−
1
2
ln
2
9
8
{\displaystyle \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
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}}
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}
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
)
=
π
2
6
{\displaystyle \operatorname {Li} _{2}(1)={\frac {{\pi }^{2}}{6}}}
Li
2
(
2
)
=
π
2
4
{\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}}
Li
2
(
−
5
−
1
2
)
=
−
π
2
10
−
ln
2
5
−
1
2
{\displaystyle \operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)=-{\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}}
=
−
π
2
10
−
arcsch
2
2
{\displaystyle =-{\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2}
Li
2
(
−
5
+
1
2
)
=
−
π
2
15
+
1
2
ln
2
5
−
1
2
{\displaystyle \operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}+1}{2}}\right)=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}}
=
−
π
2
15
+
1
2
arcsch
2
2
{\displaystyle =-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\operatorname {arcsch} ^{2}2}
Li
2
(
3
+
5
2
)
=
π
2
15
−
1
2
ln
2
5
−
1
2
{\displaystyle \operatorname {Li} _{2}\left({\frac {3+{\sqrt {5}}}{2}}\right)={\frac {{\pi }^{2}}{15}}-{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}}
=
π
2
15
−
1
2
arcsch
2
2
{\displaystyle ={\frac {{\pi }^{2}}{15}}-{\frac {1}{2}}\operatorname {arcsch} ^{2}2}
Li
2
(
5
+
1
2
)
=
π
2
10
−
ln
2
5
−
1
2
{\displaystyle \operatorname {Li} _{2}\left({\frac {{\sqrt {5}}+1}{2}}\right)={\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}}
=
π
2
10
−
arcsch
2
2
{\displaystyle ={\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2}
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. 
