# Integration in complex plane - deforming contours

1. Apr 16, 2012

### bjnartowt

1. The problem statement, all variables and given/known data

The following sum of integrals has integrals that are both integrated over straight lines in the complex plane. Deform the contours back to the origin and avoid the singularity at x = infinity to prove the integral formula,

$\int_{ - {x_0}}^\infty {(x + {x_0}){e^{ - {{(x/a)}^2}}}dx} - \int_{ + {x_0}}^\infty {(x - {x_0}){e^{ - {{(x/a)}^2}}}dx} = 2{x_0}\int_{ - \infty }^{ + \infty } {{e^{ - {{(x/a)}^2}}}dx}$

2. Relevant equations

cauchy's integral theorem?

3. The attempt at a solution

Well, since this problem came up in the larger context of a quantum scattering problem, can I count that as my attempt of the solution? See the attached pdf for the problem statement and my developing solution...

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