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
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 Incomplete gamma function
References