二重对数函数

**斯盆司函數**（英文：**Spence's function**）也叫**二重對數函數**（英文：**Dilogarithm**），最早由歐拉提出，定義如下：

　
: $\operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )$

## 恆等式

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

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

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

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

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

## 特殊值恆等式

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

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

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

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

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

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

## 特殊值

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

　
: $\operatorname {Li} _{2}(0)=0$

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

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

　
: $\operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2$

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

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

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

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

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

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

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

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

## 在粒子物理中

斯盆司函數在粒子物理領域中計算輻射校正時比較常見。此時該函數常用一個含絕對值的對數表達式定義。

　
: $\operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u}}\,du={\begin{cases}\operatorname {Li} _{2}(x),&x\leq 1;\\{\frac {\pi ^{2}}{3}}-{\frac {1}{2}}\ln ^{2}(x)-\operatorname {Li} _{2}({\frac {1}{x}}),&x>1.\end{cases}}$

## 參考文獻

* Lewin, L. Dilogarithms and associated functions. Foreword by J. C. P. Miller. London: Macdonald. 1958. MR 0105524.
* Morris, Robert. The dilogarithm function of a real argument. Math. Comp. 1979, **33** (146): 778–787. MR 0521291. doi:10.1090/S0025-5718-1979-0521291-X.
* Loxton, J. H. Special values of the dilogarithm. Acta Arith. 1984, **18** (2): 155–166 [2015-02-02]. MR 0736728. （原始內容存檔於 2015-02-02）.
* Kirillov, Anatol N. Dilogarithm identities. 1994. arXiv:hep-th/9408113 .
* Osacar, Carlos; Palacian, Jesus; Palacios, Manuel. Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astron. 1995, **62** (1): 93–98. doi:10.1007/BF00692071.
* Zagier, Don. The Dilogarithm Function (PDF). Front. Number Theory, Physics, Geom. II. 2007: 3–65 [2015-02-02]. doi:10.1007/978-3-540-30308-4\_1. （原始內容存檔 (PDF) 於 2015-03-26）.