# Gaussian convolution question

1. Sep 16, 2009

### astralmeme

Greetings,

I am a computer scientist revisiting integration after a long time. I am stuck with this simple-looking integral that's turning out to be quite painful (to me). I was wondering if one of you could help.

The goal is to solve the integral

$$\int_{0}^{\infty} e^{-(x - t)^2/2 \sigma^2} x^n\ dx .$$

Note that this is the convolution of the Gaussian centered around 0 with the function that equals $x^n$ for $x > 0$, and 0 elsewhere (modulo scaling).

In particular, I would be interested in seeing any relationship with the integral

$$\int_{-\infty}^{\infty} e^{-(x - t)^2/2 \sigma^2} x^n\ dx .$$

which I have worked out.

Any suggestions?

Thanks in advance,
Swar

2. Sep 16, 2009

### HallsofIvy

Staff Emeritus
If n is odd, that can be done by letting $u= -(x-\mu)^2/(2\sigma^2)$. If x is even, try integration by parts, letting $u= x^{n-1}$, $dv= xe^{-(x-\mu)^2/(2\sigma^2)}$ to reduce it to n odd.

3. Sep 16, 2009

### mathman

As long as t is not 0, the best you can do is express the integral in terms of the error function.

4. Sep 16, 2009

### astralmeme

Regarding error function, that is my guess too, but can you tell me what exactly would need be done?

I apologize if the question is obvious.

Best,
Swar

5. Sep 17, 2009

### mathman

What I would do is first let u=x-t. Then xn becomes (u+t)n.
Expand the polynomial in u and then by succesive itegration by parts, get all the terms to a 0 exponent for u, which will be proportional to erf(t).

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