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I'm working the following paper ""Plane waves viewed from an accelerated frame, K Srinivasan, L Sriramkumar, T Padmanabhan - Physical Review D, 1997"

and there's this integral:

[tex]

\int_{-\infty}^{+\infty} e^{- i \Omega t} \cos \left( \beta - e^{a(\phi/\Omega-t)}\right) dt

[/tex]

whose result seems to be

[tex]

= \frac{e^{- i \phi}}{2 a}\Gamma\left(\frac{i \Omega}{a}\right) \left( e^{\Omega/4\Omega_0} e^{i \beta}+ e^{-\Omega/4\Omega_0} e^{-i \beta}\right)

[/tex]

where [tex] \Omega_0 = a/2 \pi [/tex]

Following the paper I changed variable [tex]z= e^{a(\phi/\Omega -t)}[/tex]. The integral is then proportional to

[tex]

\int_{0}^{\infty} z^{\frac{i \, \, \Omega}{a} -1}\left(e^{i (\beta -z)}+e^{-i (\beta -z)}\right) dz

[/tex]

This is looking a bit a like a Gamma function. The paper then says "analytically continuing to Im z". This is not clear to me. Shall I integrate along some path in the complex plane? Which one? I tried along the first quadrant of C (between R+ and Im+), avoiding the pole in z=0, but its not clear to me how to control the contributions of the paths of very small radius of very large.

I could use some help! :)

Thanks

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# A tricky integral

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