How can the error function be used to solve this integral?

[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

 

[HallsofIvy]

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.

 

[mathman]

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

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

 

[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.


Swar

 

[mathman]

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.


Swar

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).