德拜函数

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

德拜函数

${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.}$

展开式

${\displaystyle D_{n}(x)=1-{\frac {n}{2(n+1)}}x+n\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k+n)(2k)!}}x^{2k},\quad |x|<2\pi ,\ n\geq 1}$。

其中${\displaystyle B_{2k}}$是伯努利数。

${\displaystyle D_{n}(x)={\frac {n*((-1)^{n}*n!*\zeta (n+1)+\sum _{m=0}^{n}((-1)^{n-m+1}*n!*x^{m}*Li_{n-m+1}(e^{x}/m!))}{x^{n+1}}}-{\frac {n}{n+1}}}$^[1]

其中${\displaystyle Li_{m}(x)}$是m阶多重对数

渐近式

For ${\displaystyle x\rightarrow 0}$ :

${\displaystyle D_{n}(0)=1}$。

For ${\displaystyle x\ll 1}$ : ${\displaystyle D_{n}}$:${\displaystyle D_{n}(x)\propto \int _{0}^{\infty }{\rm {d}}t{\frac {t^{n}}{\exp(t)-1}}=\Gamma (n+1)\zeta (n+1).\quad [\Re \,n>0]}$ ^[2]

相关函数

 

Debye function Maple complex3D animation

也有将德拜函数定义为^[3]

${\displaystyle d_{n}(z)=\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}dt}$  ${\displaystyle =n!*\zeta (n+1)-{\frac {x^{n+1}}{n+1}}+\sum _{k=0}^{n}((-1)^{k+1}*(\prod _{j=0}^{k-1}((n-j)*x^{n-k}*Li_{k+1}(exp(x)))))}$ 

参考文献

1. 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
2. Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series, and Products (Academic, New York, 1980), (3.411).
3. Milton abramowitz Irene Stegun, Handbook of Mathematical Functions,National Bureau of Standards, p998 1972