Complex Substitution and Infinity in Quantum Mechanics Integrals

[prismaticcore]

Homework Statement

In Griffiths' Introduction to Quantum Mechanics problem 2.22 as well as 6.7, I used substitution to complete an integral. The original integral had limits from negative infinity to positive infinity. For my substitution, I had a complex constant term added to the original variable. In computing the new limits of the integral after substitution, I must somehow add a complex number to infinity. Does this imply that the new limits are also negative infinity to infinity? Also, I haven't had analysis nor complex analysis and so I am unsure as to how to appropriately phrase what is going on in computing these new limits.

Homework Equations

The Attempt at a Solution

 

[Dick]

The new integral is over a line shifted by a complex constant from the original line. IF there are no poles in the function between the original line and new line and IF the functions are 'well behaved' at infinity, i.e. go to zero fast enough, then you can argue using the Cauchy Integral theorem that the integral over both lines are equal. Given this is a quantum mechanics problem and not a complex analysis problem I suspect both IF's are probably true. So, yes, you can do that.