斯盆司函数的二维图像
斯盆司函数在复数域上的图像
斯盆司函数(英文:Spence's function)也叫二重对数函数(英文:Dilogarithm),最早由欧拉提出,定义如下:
![{\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f028651fd71784c2c5b305f81280dbfb0236ebb3)
恒等式[编辑]
[1]
[2]
[1]
[2]
[1]
特殊值恒等式[编辑]
[2]
[2]
[2]
[2]
[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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5f124e544fede0ab8df881d796ac67c248e177)
特殊值[编辑]
![{\displaystyle \operatorname {Li} _{2}(-1)=-{\frac {{\pi }^{2}}{12}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86cfc7d8f5930c0ae03b41f2ff003b57837bc142)
![{\displaystyle \operatorname {Li} _{2}(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d336382831f60ce3508426546a0c35ab57152d78)
![{\displaystyle \operatorname {Li} _{2}\left({\frac {1}{2}}\right)={\frac {{\pi }^{2}}{12}}-{\frac {\ln ^{2}2}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0d3eb4bcdde66c9f8b9827f641d832d8b7fb2f)
![{\displaystyle \operatorname {Li} _{2}(1)={\frac {{\pi }^{2}}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298fd2ad1df8e53fcd20b2a8a56295f70ba961cf)
![{\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8994f5d34ac325ad84ca2c2d8ee347a7e869aa)
![{\displaystyle =-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\operatorname {arcsch} ^{2}2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/290d4c0086063cb4e2c7822824870f6b201a305f)
![{\displaystyle =-{\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426c8283ce300978b1ece46c20431fe8c68a5e8a)
![{\displaystyle ={\frac {{\pi }^{2}}{15}}-{\frac {1}{2}}\operatorname {arcsch} ^{2}2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4745f8c99b4f68e3227ea2c746668c11defcbab4)
![{\displaystyle ={\frac {{\pi }^{2}}{10}}-\operatorname {arcsch} ^{2}2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5830918e06d3581d7f819b95e5efb87f695296)
在粒子物理中[编辑]
斯盆司函数在粒子物理领域中计算辐射校正时比较常见。此时该函数常用一个含绝对值的对数表达式定义。
![{\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 ^{2}(x)-\operatorname {Li} _{2}({\frac {1}{x}}),&x>1.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/073b836aee4127f4036dfa6c9b70121f75b5d8fb)
参考文献[编辑]
- ^ 1.0 1.1 1.2 Zagier
- ^ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Weisstein, Eric W. (编). Dilogarithm. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语).
- 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. (原始内容存档于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. (原始内容存档 (PDF)于2015-03-26).