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]
Estructura analítica
modifica
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] Identitats de valor particular
modifica
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}.}
Valors especials
modifica
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}}}
En física de partícules
modifica
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 Bibliografia addicional
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Enllaços externs
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