How to get Mellin transform of the given function?

Question

Could anyone help me how to find Mellin transform of the following function $$f(x)=\frac{1}{(x^a-A)^3},$$ i.e. the integral $$\int_0^{+\infty} \frac{x^{s-1}\,{\rm d}x}{(x^a-A)^3}$$ where $a,A>0\,.$

Answer 1

• If $A > 0$ then it is a principal value, thus the mean value of $A+i\epsilon,A-i\epsilon$. Whence we can assume $A\not \in [0,\infty)$

• With a change of variable we get $a=1$. With another one we get $\mathcal{M}[\frac1{(x+1)^3}]$

• The residue theorem gives for $c\in (0,1),x\ne 1$ $$\frac1{x+1}=\frac1{2i\pi}\int_{c-i\infty}^{c+i\infty} \frac{\pi}{\sin(\pi s)} x^{-s}ds$$ Differentiating we get $$\frac2{(x+1)^3}=\frac1{2i\pi}\int_{c-i\infty}^{c+i\infty} \frac{\pi}{\sin(\pi s)}s(s+1) x^{-s-2}ds$$

$$\implies \mathcal{M}[\frac2{(x+1)^3}]=\frac{\pi}{\sin(\pi (s-2))}(s-2)(s-1)\qquad (\Re(s)\in (0,3))$$