Is it possible to compute the inverse transform of
$$ \frac{1}{a^{-s}\cos( \frac{\pi s}{2})\Gamma (s)} $$
or similarly is it possible to compute the Inverse Mellin transform ??
$$ \frac{ \zeta (1-s)}{\zeta (s)} $$
$$ \frac{ \zeta (s)}{\zeta (1-s)} $$
The Mellin inverse is given by
$$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $$
Best Answer
For the first one $$ \mathcal{M}^{-1}\left[ \frac{1}{a^{-s}\cos( \frac{\pi s}{2})\Gamma (s)} \right] = \frac{2 a \cos\left(\frac{a}{x}\right)}{\pi x} $$
For the $\zeta$ expressions we can use the Riemann functional equation $$ \zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s) $$ Then find the inverse Mellin transforms $$ \mathcal{M}^{-1}\left[2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\right] = \frac{2 \cos(\frac{2\pi}{x})}{x} $$ and $$ \mathcal{M}^{-1}\left[\frac{1}{2^s \pi^{s-1} \sin(\frac{\pi s}{2})\Gamma(1-s)}\right] = 2 \cos(2 \pi x) $$