Homogenous function

**champrock** · May 10th 2009, 04:02 AM

Hello

The following function is homogenous of what degree?

f(x1,x2) : Integration of e^-{w^2/(x1^2+x2^2)} dw with limits from 0 to [x1^2+x2^2]^0.5

really lost in how to integrate this function and then find the required answer.

**Danny** · May 10th 2009, 05:18 AM

Originally Posted by champrock

Hello

The following function is homogenous of what degree?

f(x1,x2) : Integration of e^-{w^2/(x1^2+x2^2)} dw with limits from 0 to [x1^2+x2^2]^0.5

really lost in how to integrate this function and then find the required answer.

A function is homogneous of degree n if $f(\lambda x) = \lambda^nf(x)$ , which in your case, because of the expontential is zero. Now for your integral what you want

$<br /> \int_0^{\sqrt{x_1^2+x_2^2}} e^{-w^2/(x_1^2+x_2^2)}dw<br />$

If you let $y = \frac{w}{\sqrt{x_2^2+x_2^2}}$ and $r = \sqrt{x_2^2+x_2^2}$ then the integral becomes

$<br /> r \int_0^1 e^{-y^2}dy = \frac{r \sqrt{\pi}}{2}\, \text{erf(1)}<br />$

**champrock** · May 10th 2009, 06:00 AM

can u please tell me what is the answer? I am getting "Homogeneous of degree 1". is that correct?

thanks

**HallsofIvy** · May 10th 2009, 06:44 AM

danny arrigo said

A function is homogeneous of degree n if $f(\lamba x)= \lambda^n f(x)$ , which in your case, because of the expontential, is zero.

(I seem to be following danny arrigo around!)

**Danny** · May 10th 2009, 07:55 AM

Sometimes people just need to hear more than once!

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