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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

$$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $$

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For the Mellin transformed function, you have to provide the strip on which it is defined; the reverse transform depends on this strip ($c$ has to within the strip). – Fabian Dec 16 '12 at 17:36
in this case , can we find a function so $$ \frac{\zeta (1-s)}{\zeta (s)}= \int_{0}^{\infty}dt f(t)t^{s-1} $$ – Jose Garcia Dec 16 '12 at 21:46
again, for which values of $\text{Re} s$ this relation should hold? – Fabian Dec 16 '12 at 22:14
Hint: $\int_0^\infty x^s\sin ax~dx=a^{-s-1}\Gamma(s+1)\cos\dfrac{\pi s}{2}$ , according to – doraemonpaul Jul 5 '15 at 0:24

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