  The variable change w=exp(u) is applied to establish novel integral
representations of the incomplete gamma-function, hypergeometric
F-function,confluent hypergeometric /Phi-function and beta-function, and to
analyze these functionsas as well as the Riemann /zeta(s,a)-function. In
particular, using these representations we give a pedagogically instructive
proof of the well known approximate functional relation for the Riemann
/zeta-function and derive Hurwitz representation of the /zeta(s,a)-function.
