Mathematics Asked on January 3, 2022

We all know that the Inverse Mellin Transform is

$$left{mathcal{M}^{-1}varphiright}(x) = f(x)=frac{1}{2 pi i} int_{c-i infty}^{c+i infty} x^{-s} varphi(s), mathrm{d}s.$$

So what is my problem? I want to calculate an integral, which looks like:

$$frac{1}{2 pi i} int_{c-i infty}^{c+i infty} x^{s} varphi(s), mathrm{d}s$$

I want to express this integral as a Inverse Mellin Transformation. My idea was to do a small substitution, so that $s rightarrow -s$. But then the integral has different boundaries:

$$-frac{1}{2 pi i} int_{-c+i infty}^{-c-i infty} x^{-s} varphi(s), mathrm{d}s = frac{1}{2 pi i} int_{-c-i infty}^{-c+i infty} x^{-s} varphi(s), mathrm{d}s$$

This is an Inverse Mellin Transform but $crightarrow-c$. How can I get to a real Inverse Mellin Transform?

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