En matemàtiques , el dilogaritme o funció d'Spence , denotada com a Li₂(z ) , és un cas particular de polilogaritme . Existeixen dues funcions especials relacionades que s'anomenen funció de Spence, el propi dilogaritme:
El dilogaritme al llarg de l'eix real
Li
2
(
z
)
=
−
∫
0
z
ln
(
1
−
u
)
u
d
u
,
z
∈
C
{\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C} }
i el seu reflex. Per a |z|<1, també es pot escriure com a sèrie infinita (la definició integral constitueix la seva extensió analítica al pla complex ):
Li
2
(
z
)
=
∑
k
=
1
∞
z
k
k
2
.
{\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.}
Alternativament, la funció de dilogaritme de vegades es defineix com a
∫
1
v
ln
t
1
−
t
d
t
=
Li
2
(
1
−
v
)
.
{\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).}
En geometria hiperbòlica , el dilogaritme es pot utilitzar per a calcular el volum d'un símplex ideal . Concretament, un símplex els vèrtexs del qual tenen una proporció creuada z
D
(
z
)
=
Im
Li
2
(
z
)
+
arg
(
1
−
z
)
log
|
z
|
.
{\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.}
La funció D (z )[1] La funció de Lobachevsky i la funció de Clausen són funcions estretament relacionades.
El dilogaritme va ser estudiat per primer cop pel matemàtic escocès de principis del segle XIX, William Spence.[2]
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}).}
[3]
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}}.}
[4]
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).}
[3]
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).}
[4]
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}}.}
[3]
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}}.}
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}}.}
[4]
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}}.}
[4]
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}.}
[4]
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}.}
[4]
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
)
=
ζ
(
2
)
=
π
2
6
,
{\displaystyle \operatorname {Li} _{2}(1)=\zeta (2)={\frac {{\pi }^{2}}{6}},}
ζ
(
s
)
{\displaystyle \zeta (s)}
funció zeta de Riemann .
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}}}
El dilogaritme apareix sovint en problemes teòrics de física de partícules en càlculs de correccions radiatives. En aquest context, la funció sovint es defineix amb un valor absolut dins del logaritme:
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}}}
↑ Zagier p. 10
↑ «William Spence - Biography ». ↑ 3,0 3,1 3,2 Zagier
↑ 4,0 4,1 4,2 4,3 4,4 4,5 Weisstein , Eric W.«Dilogarithm» a MathWorld (en anglès).
Lewin , L.; Foreword by J. C. P. Miller. Dilogarithms and associated functions . Londres: Macdonald, 1958. Morris , Robert Math. Comp. , 33, 1979, pàg. 778–787. DOI : 10.1090/S0025-5718-1979-0521291-X [Consulta: free].Loxton , J. H. Acta Arith. , 18, 1984, pàg. 155–166. DOI : 10.4064/aa-43-2-155-166 [Consulta: free].Kirillov , Anatol N. Progress of Theoretical Physics Supplement , 118, 1995, pàg. 61–142. arXiv : hep-th/9408113 . Bibcode : 1995PThPS.118...61K . DOI : 10.1143/PTPS.118.61 .Osacar , Carlos; Palacian , Jesus; Palacios , Manuel Celest. Mech. Dyn. Astron. , 62, 1995, pàg. 93–98. Bibcode : 1995CeMDA..62...93O . DOI : 10.1007/BF00692071 .Zagier , Don. «The Dilogarithm Function». A: Pierre Cartier . Frontiers in Number Theory, Physics, and Geometry II DOI 10.1007/978-3-540-30308-4_1 ISBN 978-3-540-30308-4 
