How to do this integral with substitution?

[rwooduk]

This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it

$$\frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq$$

The only information the tutor has given use to solve this is to use substitution and this:

$$\int_{\infty }^{\infty} e^{\tfrac{-q^{2}}{2\Delta^{2}}} dq = \sqrt{2 \pi}\Delta$$

Please could someone give me a point int the right direction? If w is the new variable what should i put it equal to?

Thanks for any help!

 

[SteamKing]

This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it

$$\frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq$$

The only information the tutor has given use to solve this is to use substitution and this:

$$\int_{\infty }^{\infty} e^{\tfrac{-q^{2}}{2\Delta^{2}}} dq = \sqrt{2 \pi}\Delta$$

Please could someone give me a point int the right direction? If w is the new variable what should i put it equal to?

Thanks for any help!

These integrals do not have solutions composed of elementary functions, and hence are not amenable to solution by substitution, except to get them into one of the integral forms discussed in the articles below.

They are related to the normal probability distribution and thus have been studied extensively analytically and numerically.

You might want to consult these articles:

http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions

 

[Mark44]

This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it

$$\frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq$$

Are you missing a sign on the lower limit of integration?
Also, what is $\Delta$? Since the variable of integration is q, then presumably $\Delta$ and x are to be treated as constants.

What do you mean by "squared bracked"? Some people call these -- [ ] -- square brackets. Did you mean that the quantity in brackets is squared, like this -- [ ... ]²?

 

[mathman]

The exponent in the original integral looks completely screwed up. Rewrite it.

 

[zoki85]

People won't stop bombarding this forum with nonelementary integrals, won't they? :D