Help with integral of a gaussian function

[maxtor101]

Hi all!

I'm curious as to how one would go about actually computing this integral

$$\int_0^\pi \exp(\frac{-x^2}{2c^2})\sin(\frac{m\pi x}{2}) dx$$

I start off by using integration by parts but I am unsure how to solve this integral

$$v = \int_0^\pi \exp(\frac{-x^2}{2c^2}) dx$$

Any help would be greatly appreciated
Max

 

[grzz]

Note that $\int$exp(-ax$^{2}$)dx x $\int$exp(-ay$^{2}$)dy = $\int$$\int$exp{-a[x$^{2}$+y$^{2}$]}dxdy.

Then change to polar coordinates.

 

[Number Nine]

Note that a lot of integrals of that form (including the normal probability density function) don't have solutions in closed form.

 

[Mute]

Note that $\int$exp(-ax$^{2}$)dx x $\int$exp(-ay$^{2}$)dy = $\int$$\int$exp{-a[x$^{2}$+y$^{2}$]}dxdy.

Then change to polar coordinates.

I don't think the OP is doing the integral you think he's doing.

Hi all!

I'm curious as to how one would go about actually computing this integral

$$\int_0^\pi \exp(\frac{-x^2}{2c^2})\sin(\frac{m\pi x}{2}) dx$$

I start off by using integration by parts but I am unsure how to solve this integral

$$v = \int_0^\pi \exp(\frac{-x^2}{2c^2}) dx$$

Any help would be greatly appreciated
Max

You're not really going to have much luck with that integral. Without the sine, the integral can only be expressed in terms of the error function, but the error function is defined by

$$\mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x dt~e^{-t^2},$$

so you haven't gained much except that you know you can express it as the error function, which mathematica, matlab, etc., have code to compute.

If you through the sine back in there, that just makes it worse. wolframalpha calculates the integral in terms of functions related to the error function. I imagine the way it does this is to use Euler's identity and write

$$\sin x = \frac{e^{ix}-e^{-ix}}{2i},$$

and then complete the square in the exponential, but there's some trickiness associated with that because you'll be introducing imaginary numbers into the limits of the integral. That's not really a problem if you know contour integrals and complex analysis, but if you don't writing the integral that way isn't really going to help you much.

 

[CellCoree]

Hello Max,

Thank you for reaching out for help with this integral. The integral you have provided is known as the Gaussian integral and it is commonly used in probability and statistics. It can be a bit tricky to solve, but there are a few different methods you can use to compute it. One approach is to use the substitution u = x/c, which will simplify the integral to:

v = \int_0^\pi \exp(\frac{-u^2}{2}) du

This integral can then be solved using techniques from calculus, such as integration by parts or the substitution method. Another approach is to use a series expansion for the exponential term and then integrate each term separately. This can be a bit more tedious, but it can also be a good exercise in using series expansions.

I hope this helps and good luck with solving the integral! Remember to always check your work and make sure your answer makes sense in the context of the problem. Let me know if you have any further questions or if you need any clarification. Happy computing!

Best,