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How to prove Fourier transformation rules of the following function

\int k^a * exp[i*k*x] * dk = x^{-1-a} * Gamma[1+a] * sin[a*pi/2] ------------(1)

I need this equation to prove some conclusions in translating propagators

in momentum space to position-space. I wish those know solutions to this

question give me hints or references urgently.

The following is the information I know, but I cannot get to the final goal.

By variable changes, k*x -> z, it's easy to get the factor k^{-1-a}, i.e.

l.h.s -> (\int z^a * exp[i*z] * dz) / x^{1+a} -------------------------------------(2)

but the remaining integration seems very difficult.

We know,

\int z^a * exp[-z] * dz \propto Gamma[1+a] --------------------------------------(3)

So, my key question is, how to do integrations in eq(2) whose exponential

argument is imaginary instead negative?

Thanks to everyone!

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# A question related with propagators in position space

Can you offer guidance or do you also need help?

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