I should prove the Frullani integral equality
∫∞0(1−ezx)βxe−γxdx=βlog(1−zγ)
for z∈C with non-positive real part.
I should first consider z≤0 and use
e−γx−e−(γ−z)xx=∫γ−zγe−yxdy
and then change the order of integration. These steps are clear (see also Frullani integral for f(x)=e−λx).
But then I should use analytic extension to show that the formula is valid for z∈C with non-positive real part. I need help for this step.
Thank you in advance!
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