Logarithmic expansion of Ramanujan's function ($N(t)\sim {\sum }_{n\ge 0}{c}_n(\log t{)}^{-n-1},{\sum }_{n\ge 0}{c}_n{z}^n/n!=1/\Gamma (1-z)$)

The Gamma reflection formula gives $(\sin \pi z/\pi )\Gamma (z)=1/\Gamma (1-z)$, analytic at zero with value one. A finite-parameter Watson's lemma representation of Ramanujan's function therefore gives the displayed coefficients. In particular $c_0=1$, $c_1=-\gamma$ and $c_2={\gamma }^2-{\pi }^2/6$, with $\gamma$ Euler's constant. Obtaining this representation requires taking an imaginary part before sending its Mellin exponent to zero; the original complex integral is divergent at that boundary.