In mathematics and statistical physics, the transport integrals (sometimes transport functions) are a family of special functions arising in the theory of transport phenomena of solids.
Definition
The transport integrals are defined by the integral representation[1][2]
-
J
n
(
x
)
=
∫
0
x
t
n
e
t
(
e
t
−
1
)
2
d
t
.
{\displaystyle J_{n}(x)=\int _{0}^{x}t^{n}{\frac {e^{t}}{(e^{t}-1)^{2}}}\,dt.}
Note that
-
e
t
(
e
t
−
1
)
2
=
∑
k
=
0
∞
k
e
k
t
.
{\displaystyle {\frac {e^{t}}{(e^{t}-1)^{2}}}=\sum _{k=0}^{\infty }k\,e^{kt}.}
See also
References
- Rogers, William; Powell, Robert (July 3, 1958). Tables of transport integrals (PDF). National Bureau of Standards. Retrieved 16 March 2026.
- Galassi, Mark. GNU Scientific Library Reference Manual (PDF) (3 ed.). ISBN 0954612078. Retrieved 16 March 2026.