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In mathematics and the field of transportation theory, the transport functionscite-ref-1[1] J(n,x) are defined by

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

• Incomplete gamma function

cite-note-11. ↑ "transport in nLab". ncatlab.org. Retrieved 14 July 2025.