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
D
n
(
x
)
=
n
x
n
∫
0
x
t
n
e
t
−
1
d
t
.
{\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.}
展开式
D
n
(
x
)
=
1
−
n
2
(
n
+
1
)
x
+
n
∑
k
=
1
∞
B
2
k
(
2
k
+
n
)
(
2
k
)
!
x
2
k
,
|
x
|
<
2
π
,
n
≥
1
{\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
B
2
k
{\displaystyle B_{2k}}
伯努利数 。
D
n
(
x
)
=
n
∗
(
(
−
1
)
n
∗
n
!
∗
ζ
(
n
+
1
)
+
∑
m
=
0
n
(
(
−
1
)
n
−
m
+
1
∗
n
!
∗
x
m
∗
L
i
n
−
m
+
1
(
e
x
/
m
!
)
)
x
n
+
1
−
n
n
+
1
{\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
L
i
m
(
x
)
{\displaystyle Li_{m}(x)}
多重对数
渐近式 For
x
→
0
{\displaystyle x\rightarrow 0}
D
n
(
0
)
=
1
{\displaystyle D_{n}(0)=1}
For
x
≪
1
{\displaystyle x\ll 1}
D
n
{\displaystyle D_{n}}
D
n
(
x
)
∝
∫
0
∞
d
t
t
n
exp
(
t
)
−
1
=
Γ
(
n
+
1
)
ζ
(
n
+
1
)
.
[
ℜ
n
>
0
]
{\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]
Debye function Maple complex3D animation 也有将德拜函数定义为[ 3]
d
n
(
z
)
=
∫
0
x
t
n
e
t
−
1
d
t
{\displaystyle d_{n}(z)=\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}dt}
=
n
!
∗
ζ
(
n
+
1
)
−
x
n
+
1
n
+
1
+
∑
k
=
0
n
(
(
−
1
)
k
+
1
∗
(
∏
j
=
0
k
−
1
(
(
n
−
j
)
∗
x
n
−
k
∗
L
i
k
+
1
(
e
x
p
(
x
)
)
)
)
)
{\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)))))}
^ 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
^ Gradshteyn, I. S., & Ryzhik, I. M. (1980). Table of integrals. Series, and Products (Academic, New York, 1980), (3.411).
^ Milton abramowitz Irene Stegun, Handbook of Mathematical Functions,National Bureau of Standards, p998 1972
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45584987608b3bd042c1fef7761ca01347aac77b)
