How can the integral \phi(x,t) be solved analytically?

[autobot.d]

Homework Statement

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

Homework Equations

Solve for \phi analytically

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

 

[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 )

 

[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.

 

[autobot.d]

This to me looks like fresnel integral.

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

Using this I get

2\sqrt{\frac{\pi}{8}}\left(1-i\right)

Any thoughts?

 

[tiny-tim]

that wikipedia link mentions the contour integral proof

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