Pi Function

see math notation, integral, euler's constant

notation PP z

definition given a CC z with z^re > 0,

PP z = $ t[z] -- ee[t] | dd t {@@ . 0}

note the pi function is defined for all CC z such that +NN (.z . 1) (it is not defined for negative integers but is defined for 0).

the gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles --- https://en.wikipedia.org/wiki/Gamma_function

theorem given a NN z,

PP z = 1 | ... z