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For what values of [itex]\epsilon[/itex] are these integrals convergent?

Would it be [itex]\epsilon \geq 0[/itex]?

Then I'm asked to use [itex]x^{-\lambda} \Gamma(\lambda) = \int_0^\infty d \alpha \alpha^{\lambda-1}e^{-\alpha x}[/itex] for [itex]x>0[/itex] and the identity [itex]\Gamma(1-\epsilon) \Gamma(\epsilon)=\frac{\pi}{\sin{\pi \epsilon}}[/itex] to show

[itex]I(c)=-\frac{\pi}{\sin{\pi \epsilon}}c^{1-\epsilon}[/itex]

I am really struggling to rewrite that original integral in terms of Gamma functions. Clearly to get the result we want, we want to show that I can be written in terms of the product of those two Gamma functions. But note that if we expand [itex]\Gamma(1-\epsilon)\Gamma(\epsilon)[/itex] we get a double integral whereas the expression we are given for [itex]I(c)[/itex] is just a single integral. How do I get this to work?

Thanks.