I have a Bessel function
$ x^{2}J''+xJ' + (x^{2}+m^{2})J=0 $
Supposing $ J(x) = x^{m}j(x) $ the equation can be reduced to $$ x(j'' + j) + (2m+1)j'=0 $$
My question is, how do i show that $$ (2m-1)z\tilde{j}= (z^{2} +1 )\frac{d\tilde{j}}{dz}$$
by using the contour integral
$$ j(x)= \lmoustache e^{zx}\tilde{j(z)} dz $$