Transport integrals

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.

The transport integrals are defined by the integral representation¹²

${\displaystyle J_{n}(x)=\int _{0}^{x}t^{n}{\frac {e^{t}}{(e^{t}-1)^{2}}}\,dt.}$

Note that

${\displaystyle {\frac {e^{t}}{(e^{t}-1)^{2}}}=\sum _{k=0}^{\infty }k\,e^{kt}.}$