Residue Convolution

We consider a residue convolution operation INTRODUCTION Consider the definition of a residue $$ Res(f,s) = (z-s)f(z) $$ for a function f with a pole at s. Here the role of (z − s) is to neutralise the infinity at f(z) with a zero. Consider this to actually be a function g(x)=x, and extend the definition to $$ Res(g,f,s) = g(z)f(z) $$ for any function which has a zero at g(s). This could be seen as a form of convolution of the two functions. If multiple residues are to be summed over, it would be interesting to consider functions with coinciding zeros and infinities. For example $$ Res(\sin(\pi \cdot),\Gamma(\cdot),s) = \sin(\pi z)\Gamma(z) $$ The sum over residues is $$ ^\infty {k!} = \pi e $$ for Res(sin²(π ⋅ ), Γ(⋅),s)² we have $$ ^\infty {k!^2} = \pi^2 I_0(2) $$ We can define this into a transform which defines a function like an inverse Mellin transform For g = sin(πx)cos(πx), f = Γ(s), we get πe−x. g f ℳ−1 ---------------- ------- ------------------------- sin(πx) Γ(s) πex sin²(πx) Γ(s)² $\pi^2 I_0(2 )$ sin(πx)cos(πx) Γ(s) πe−x