# Unable to figure out integral

## Homework Statement

I have an integral in the form of
[/B]
$$\int_{0}^{\infty} e^{-(\frac{x-c}{b})^2}dx$$

and I can't seem to figure it out.

I feel like I need a special function (gamma?)

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## Answers and Replies

Ooops I think I realized I need to do a change of variables...

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pasmith
Homework Helper
The LaTeX for infinity is \infty, which produces $\infty$.

To your problem: the square of that integral is $$\int_0^\infty \int_0^\infty e^{-(\frac{x-c}b)^2 - (\frac{y-c}b)^2}\,dx\,dy$$ and the substitution $$x - c = r \cos \theta \\ y - c = r \sin \theta$$ is called for.

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• Eats Dirt
would this work?

use change of variables

$$y=(\frac{x-c}{b})\\ dy=\frac{dx}{b}\\ substitute\\\\ \int_{0}^{\infty}e^{-ay^2}dy = \frac{1}{2}\sqrt{\frac{pi}{a}}$$

oops I did not make up for the change of variables in the lower limit of the integral

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vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
Almost. You need to fix the limits of integration.

$$\int_{-c/b}^{\infty}e^{-ay^2}dy =! \frac{1}{2}\sqrt{\frac{pi}{a}}$$

Now the Gaussian doesn't work in that form so I think I need the indefinite form Last edited:
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
$$\int_{-c/b}^{\infty}e^{-ay^2}dy =! \frac{1}{2}\sqrt{\frac{pi}{a}}$$

Now the Gaussian doesn't work in that form so I think I need the indefinite form and I just evaluate this in my new limits?

You tell us.

RUber
Homework Helper
You could also just use some brute force and integration by parts:
let ##\alpha = \frac{x-c}{b}##
You will have ## \int_{\frac{-c}{b}}^\infty e^{-\alpha^2} d\alpha ##
by parts, ##u = e^{-\alpha^2}, \, du = -2\alpha u d\alpha ##
## v = \alpha, \, dv = d\alpha ##
## \int_{\frac{-c}{b}}^\infty e^{-\alpha^2}d\alpha = \left. uv \right|_{\frac{-c}{b}}^\infty + 2 \int_{\frac{-c}{b}}^\infty \alpha^2 e^{-\alpha^2} d\alpha ##
##\int_{\frac{-c}{b}}^\infty \alpha^2 e^{-\alpha^2} d\alpha ## can again be evaluated by parts...
substituting ##\beta = \alpha^2 ## gives ##-\int_{\frac{-c^2}{b^2}}^{-\infty} \beta e^{\beta} d\beta ##
Let ## u = \beta, \, du = d\beta ## and ## v=e^\beta, \, dv = e^\beta d\beta ## .
One more time, and keep track of your signs, do some algebra, and before you know it... Solved.
However, I think the suggesions above may prove to be more elegant.

You tell us.

Ok, now we use the error function,
$$erf(y)=\frac{2}{\sqrt{\pi}}\int_{0}^{y} e^-t^2 dt$$

After I evaluate the error function I should put in the proper integration limits (from the original function) of y? But doesn't this just lead us in circles now? to evaluate this function I need another Gaussian function in a form that does not run from 0 to infinity.

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
Ok, now we use the error function,
$$erf(y)=\frac{2}{\sqrt{\pi}}\int_{0}^{y} e^-t^2 dt$$

After I evaluate the error function I should put in the proper integration limits (from the original function) of y? But doesn't this just lead us in circles now? to evaluate this function I need another Gaussian function in a form that does not run from 0 to infinity.

No, you are not running in circles. You are expressing an integral (the one you started with) in terms of a function "erf" (or equivalent). The latter is not an elementary function, but it is well-studied and has a standard definition.

The only real difference between a function like "erf" and functions like "sin", "cos", "exp" "log", etc, is that you saw these last ones early in your education and so you are more familiar with them and more practiced in their use. Also, you can probably find them on just about any decent hand-held calculator, while "erf" shows up later in your studies and needs a fancier calculator to implement it.

pasmith
Homework Helper
The LaTeX for infinity is \infty, which produces $\infty$.

To your problem: the square of that integral is $$\int_0^\infty \int_0^\infty e^{-(\frac{x-c}b)^2 - (\frac{y-c}b)^2}\,dx\,dy$$ and the substitution $$x - c = r \cos \theta \\ y - c = r \sin \theta$$ is called for.

Actually the $\theta$ intergral here is not straightforward due to the nature of the boundary, so using the error function is easier.

RUber
Homework Helper
##\int_{-c/b}^\infty e^{-t^2} dt ## has the form of the complimentary error function ##1-erf(x)##, as described on the Wikipedia page linked above.
You can also read more at http://www.wolframalpha.com/input/?i=erfc(x) .
If you would prefer the expansion in terms of the given values without functional dependence on the erf, you can go through the somewhat messy integration I outlined above.

thank you all, I ended up getting the correct solution.