Thursday, September 20, 2018

calculus - Frullani integral for f(x)=ex in a complex context

I should prove the Frullani integral equality



0(1ezx)βxeγxdx=βlog(1zγ)


for zC with non-positive real part.



I should first consider z0 and use
eγxe(γz)xx=γzγeyxdy


and then change the order of integration. These steps are clear (see also Frullani integral for f(x)=eλx).



But then I should use analytic extension to show that the formula is valid for zC with non-positive real part. I need help for this step.



Thank you in advance!

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