Integral arising from the inverse Fourier Transform

[Hardflip]

Homework Statement

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I was using the Fourier transform to solve the following IVP:

$\frac{\partial^2 u}{\partial t \partial x} = \frac{\partial^3u}{\partial x^3} \\ u(x,0)=e^{-|x|}$

Homework Equations

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$f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega x}d\omega$

$\hat{f}(e^{-|x|}) = \sqrt{\frac{2}{\pi}}\frac{1}{1+\omega^2}$

The Attempt at a Solution

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After working out the PDE, I've gotten to the solution:

$u(x,t) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{e^{-\omega^2 t}}{1+\omega^2}e^{i\omega x}d\omega$

And, after doing the usual trick of completing the square, I've gotten the integral down to

$u(x,t) = \frac{1}{\pi}e^{-\frac{x^2}{4t}}\int_{-\infty}^{\infty}\frac{e^{-t(\omega-\frac{ix}{2t})^2}}{1+\omega^2}d\omega$

But from here I have no idea how to proceed. Keep in mind that my professor often assigns problems by taking them from the textbook and tweaking random things without working through them himself. In a few of his assignments, he has made the problems unsolvable by doing so. If this integral has a closed form solution, I would really appreciate a hint on where to proceed from here. If it doesn't, then I would appreciate knowing just as much so I can bring it to my professor's attention.

 

[vela]

I didn't check that your solution so far is correct, though it looks like you messed up in completing the square. Your last expression doesn't look well defined for t=0.

But assuming you end up with a similar integral, contour integration in the complex plane looks like a way to go.