查看“德拜函数”的源代码

{{校对翻译}}
'''德拜函数'''（Debye function）是[[彼得·德拜]]于1912年估算[[声子]]对固体的[[比热]]的[[德拜模型]]时创立的函数，定义如下
[[File:Debye function animation.gif|thumb|300px|德拜函数]]

:<math>D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^t - 1}\,dt.</math>

;展开式
<math>D_n(x) = 1 - \frac{n}{2(n+1)} x +  n \sum_{k=1}^\infty \frac{B_{2k}}{(2k+n)(2k)!} x^{2k}, \quad |x| < 2\pi,\ n \ge 1</math>。

其中<math>B_{2k}</math>是[[伯努利数]]。

<math>D_n(x)=\frac{n*((-1)^n*n!*\zeta(n+1)+\sum_{m=0}^{n}((-1)^{n-m+1}*n!*x^m*Li_{n-m+1}(e^{x}/m!))}{x^{n+1}}-\frac{n}{n+1}</math><ref>A. E. Dubinov,  A. A. Dubinova ,Exact integral-free expressions for the integral Debye functions,Technical Physics Letters,December 2008, Volume 34, Issue 12, pp 999-1001</ref>

其中<math>Li_{m}(x)</math>是m阶[[多重对数]]

;渐近式

For <math>x \rightarrow 0</math> :
:<math>D_n(0)=1</math>。

For <math>x \ll 1</math> : <math>D_n</math>:<math>D_n(x)\propto\int_0^\infty{\rm d}t\frac{t^{n}}{\exp (t)-1} = \Gamma(n + 1) \zeta(n + 1).    \quad [\Re \, n > 0]</math> <ref>Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series, and Products (Academic, New York, 1980), (3.411).</ref>

==相关函数==
[[File:Debye function Maple complex plot.gif|thumb|300px|Debye function  Maple complex3D  animation]]
也有将德拜函数定义为<ref>Milton abramowitz Irene Stegun, Handbook of Mathematical Functions,National Bureau of Standards, p998  1972</ref>

<math>d_{n}(z)=\int_{0}^{x}\frac{t^n}{e^{t}-1}dt</math>
<math>=n!*\zeta(n+1)-\frac{x^{n+1}}{n+1}+\sum_{k=0}^{n}((-1)^{k+1}*(\prod_{j=0}^{k-1}((n-j)*x^{n-k}*Li_{k+1}(exp(x)))))</math>

==参考文献==
<references/>

[[Category:特殊函数]]