Solve ∫xp e-x2 eix dx | Get Hints for Integration

[Ananthan9470]

I need to evaluate ∫x^p e^-x2 e^ix dx from -∞ to ∞ Can someone please give me some pointers on how to do this? I am completely lost. I just need some hints or something.

 

[PeroK]

I need to evaluate ∫x^p e^-x2 e^ix dx from -∞ to ∞ Can someone please give me some pointers on how to do this? I am completely lost. I just need some hints or something.

Is this homework?

 

[Ananthan9470]

Is this homework?

No. I'm trying to learn quantum mechanics and this thing keeps popping up.

 

[PeroK]

You'll get a better response if you post it in homework. Even if you're learning on your own, it still counts.

Can you integrate it without the complex exponential?

 

[mathman]

To get you started, if p is even you need only cosx. If p is odd you need only isinx, where $e^{ix}=cosx+isinx$.
Next integrate by parts to reduce exponent from p to p-1, and continue until you get p = 0.
At the end you should have $\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}cosxdx$.

As an afterthought, it might be easier to start from
$\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}cosxdx$. Then integrate by parts to increase the exponent of x.