# Homework Help: Solving a difficult integral

1. Sep 6, 2011

### autobot.d

1. The problem statement, all variables and given/known data

$\phi\left(x,t\right)=\frac{1}{2\pi}\int^{\infty}_{-\infty}e^\left(i\left(xk-tk^2\right)\right)dk$

2. Relevant equations
Solve for $\phi$ analytically

3. The attempt at a solution
completing the square of the exponent to give me

$\phi\left(x,t\right)=\frac{1}{2\pi}\int^{\infty}_{-\infty}e^\left(-ti\left(k^2-\frac{x}{t}k + \frac{x^2}{4t^2} - \frac{x^2}{4t^2}\right)\right)dk$

Simplifying I get
$\phi\left(x,t\right)=\frac{e^\frac{x^2}{4t}}{2\pi}\int^{\infty}_{-\infty}e^\left(-ti\left(k-\frac{x}{2t}\right)^2\right)dk$

From here I don't know

tried u substitution

$u=k-\frac{x}{2t} , du=dk$
but this gets me nowhere
any help is appreciated

2. Sep 6, 2011

### tiny-tim

hi autobot.d!

there's a standard way of solving ∫-∞ e-u2 du, which you need to be familiar with …

it's something like √π (i forget exactly )

3. Sep 7, 2011

### autobot.d

The problem is that there is an i in there

$\int^{\infty}_{-\infty}e^\left(-\mathbf{i} tu^2\right) du$

The i is what I am having the problem with.

Thanks for the help.

Last edited: Sep 7, 2011
4. Sep 8, 2011

5. Sep 8, 2011

### tiny-tim

that wikipedia link mentions the contour integral proof

a detailed version is at http://planetmath.org/encyclopedia/FresnelFormulae.html" [Broken]

Last edited by a moderator: May 5, 2017