Transport integrals

The transport integrals are defined by the integral representation^[1][2]

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

Note that

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

• Incomplete gamma function