# Gaussian Integral

## Homework Statement

Consider the gaussian distribution shown below

$$\rho (x) = Ae^{-\lambda (x-a)^2$$

where A, a, and $\lambda$ are positive real constants. Use $\int^{-\infty}_{+\infty} \rho (x) \,dx = 1$ to determine A. (Look up any integrals you need)

## Homework Equations

Given in question above

## The Attempt at a Solution

My plan was to integrate the probability density set it equal to one and then solve for A. The problem is I'm getting stuck on the integration. I started by pulling the constants out of the integral and doing the substitution $u=x-a$ that left me with
$$Ae^{-\lambda} \int^{+\infty}_{-\infty} e^{u^2}\,du$$
It's been a while since calc II and I can't figure out how to do this one (even though it looks so simple). I also tried looking it up in a integral table but couldn't find it. Any help would be appreciated.

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Cyosis
Homework Helper

$$e^{-\lambda u^2} \neq e^{-\lambda}e^u^2=e^{-\lambda+u^2}$$

tiny-tim
Homework Helper
… (Look up any integrals you need) …

I also tried looking it up in a integral table but couldn't find it. Any help would be appreciated.
Hi DukeLuke!

You need the erf(x) function … see http://en.wikipedia.org/wiki/Error_function

(btw, there is a way to integrate ∫e-u2du: it's √(∫e-u2du)∫e-v2dv), then change to polar coordinates )

$$e^{-\lambda u^2} \neq e^{-\lambda}e^u^2=e^{-\lambda+u^2}$$
Thanks, man am I getting rusty over the summer

You need the erf(x) function
I looked at it but I'm lost on how to use it solve this problem. Could you help me get started?

D H
Staff Emeritus
(Look up any integrals you need)
Have you tried this bit of advise? You even know the relevant keywords (hint: use the title of this thread). Google is your friend.

$$\int_{-\infty}^{\infty} e^{-(x+b)^2/c^2}\,dx=|c| \sqrt{\pi}$$

Thanks, using the integral above from Wikipedia $c = \frac{1}{\sqrt{\lambda}}$. From there I get $A = \frac{\sqrt{\lambda}}{\sqrt{\pi}}$.

Looks correct, studying griffiths' quantum mechanics I see :D

Yep, thought I would get a head start before the fall semester begins.