Sunday, February 28, 2016

complex numbers - Method of pole shifting (feyman's trick) in Scattering theory vs contour deformation trick


I am studying Scattering theory but I am stuck at this point on evaluating this integral


$G(R)={1\over {4\pi^2 i R }}{\int_0^{\infty} } {q\over{k^2-q^2}}\Biggr(e^{iqR}-e^{-iqR} \Biggl)dq$



Where $ R=|r-r'|$


This integral can be rewritten as


$G(R)={1\over {4{\pi}^2 i R }}{\int_{-\infty}^{\infty} } {q\over{k^2}-{q^2}}{e^{iqR}}dq$


Zettili did this integral by the method of contour integration in his book of 'Quantum Mechanics'.He uses residue theorems and arrived at these results.


$G_+(R)={ -1e^{ikR}\over {4 \pi R}}$ and $G_-(R)={ -1e^{-ikR}\over {4 \pi R}}$


I don't get how he arrived at this result. The test book doesn't provide any detailed explanations about this. But I know to evaluate this integral by pole shifting.


My question is how to evaluate this integral buy just deform the contour in complex plane instead of shifting the poles?




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