Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$ where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
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Look at http://math.stackexchange.com/questions/285348/integration-fourier-transform – superAnnoyingUser Feb 16 '13 at 21:02
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1and http://en.wikipedia.org/wiki/Fresnel_integral – superAnnoyingUser Feb 16 '13 at 21:05
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Consider an integral in the complex plane along the contour which goes from the origin out along the real axis to $+R$, then moves in a circular arc up to the point $+iR$, and then returns to the origin along the imaginary axis.
Jonathan
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Or go up to $\frac {(R,R)} {\sqrt 2}$ and back to the origin. Shouldn't that work @Jonathan? – superAnnoyingUser Feb 16 '13 at 21:09
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That contour would not work with the integrand I have in mind (which is the natural one for this problem), but perhaps with a different integrand. – Jonathan Feb 16 '13 at 21:14