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

[Kate2010]

Homework Statement

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

Homework Equations

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?

 

[cragar]

square the integral and do it in polar. then take the square root of the answer when your done.
http://en.wikipedia.org/wiki/Gaussian_integral
look about half way down the page, they show you the trick

 

[jackmell]

Homework Statement

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

Homework Equations

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?

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 $\sqrt{\pi}$