Problem with a gaussian integral.

1. May 30, 2005

misogynisticfeminist

Hey, i've been learning about gaussian integrals lately. And i'm now stuck in one part. I am now trying to derive some kind of general formula for gaussian integrals

$$\int x^n e^{-\alpha x^2}$$

for the case where n is even. So they ask me to evaluate the special case n=0 and alpha=1. So its $$I= \int^{\infty}_{-\infty} e^{-x^2} dx$$. When i square this integral, they said that its $$I^2= ({\int^{\infty}_{-\infty} e^{-x^2} dx})({\int^{\infty}_{-\infty} e^{-y^2} dy})$$ with both x and y as according to them, i have to use a different variable for the first and second integral factors.

Why is this so? I have limited calc background. So i was wondering you guys could help me out. Thanks alot.....

Last edited: May 30, 2005
2. May 30, 2005

inha

I think they're trying to get you to use the gamma function.

$$\Gamma(z)=2{\int^{\infty}_{0}dte^{-t^2}t^{z-1}$$

Those two integrands you have are even and the gamma func has useful properities for situations like this.

3. May 30, 2005

Dr Transport

By using different variables for the squared version, you can convert to different coordinate systems, i.e. to cylindrical coordinates. What you typed above is the way you prove that $$\int^{\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$$

4. May 30, 2005

HallsofIvy

"When i square this integral, they said that its $$I^2= ({\int^{\infty}_{-\infty} e^{-x^2} dx})({\int^{\infty}_{-\infty} e^{-y^2} dy})$$"

No, they didn't say that! You put in the "when I square this integral" yourself- they didn't square the integral.

What they did say was: If $$I= \int^{\infty}_{-\infty} e^{-x^2} dx$$, then it is also true that $$I= \int^{\infty}_{-\infty} e^{-y^2} dy$$ because that is just a change in the dummy variable.

It is then true that $$I^2= I*I= $$\int^{\infty}_{-\infty} e^{-x^2} dx$$$$\int^{\infty}_{-\infty} e^{-y^2} dy$$$$- just multiplying two different ways of writing the same thing.

Of course, the really important thing is that fact that that product of integrals can be written as a double integral:
$$(\int^{\infty}_{-\infty} e^{-x^2} dx)(\int^{\infty}_{-\infty} e^{-y^2} dy= \int_{y= -\infty}^{\infty}\int_{x=-\infty}^{\infnty}e^{-(x^2+ y^2)}dxdy$$.

Last edited by a moderator: May 30, 2005
5. May 30, 2005

misogynisticfeminist

ahhhh so i see, a double integral. Thanks alot. Maybe i misinterpreted what they said, cos' they made it sound that changing the dummy variable was a must.