Homework Help: Integrate exp(-(z-ia)^2) from z = - infinity to z = infinity

1. Mar 17, 2012

Kate2010

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

Prove that $\int^{∞}_{-∞}$ exp(-(z-ia)2)dz = √∏ for all real a.

2. Relevant equations

3. The attempt at a solution

If I use the substitution x = z-ia then dz = dx and if I use the limits x = -∞ to x = ∞ I get the correct answer. However, I do not know how to justify leaving the limits the same or if it is even ok?

2. Mar 17, 2012

cragar

3. Mar 17, 2012

jackmell

No it's not ok but you can still solve it by making that substitution $u=z-ia$, just change the limits on the integral:

$$\int_{-\infty}^{\infty}e^{-(z-ia)^2}dz=\int_{-\infty-ia}^{\infty+ia} e^{-u^2}du$$

Now, that integrand is analytic so independent of path so that I can go from $-\infty-ia$ up to the point $-\infty$, go down the real axis to $\infty$ then up to the point $\infty+ia$. The two vertical legs are zero because of the negative exponent so that we're left with just the ordinary gaussian integral which is $\sqrt{\pi}$

Last edited: Mar 17, 2012