Efficient Integration Techniques: Solving the Gaussian Distribution

[Bladibla]

Hi everyone,

I am new to Physics Forums so please excuse me - I am so embarrassed that I can't do this integral, but its quite urgent and so if anyone here could help me I would be much obliged!

The integral I must carry out is:

$$\int_{-\infty}^{\infty}e^{-ax^{2}}\,\text{cos}kx\,dx$$

I already know that the solution is

$$\sqrt{\frac{\pi}{a}}e^{-k^{2}/4a}$$

But the task is to find out how one can get to this answer...I think the hint is in:

$$\int_{-\infty}^{\infty}e^{-ax^{2}}\,dx =\sqrt{\frac{\pi}{a}}$$

which is from the standard Gaussian distribution.

Any information would be much appreciated!

Thank you,
Bladibla.

 

[zhermes]

Try integration by parts, twice.

 

[Bladibla]

Hi zhermes, could you please be a little more specific? Thanks!

 

[Dick]

Try integration by parts, twice.

That's not it. What you want to do is write cos(kx)=(e^(ikx)+e^(-ikx))/2. Combine the exponentials. You get integrands like exp(-a*(x^2+i*x*k/a)), right? You can complete the square and change variables.

 

[Bladibla]

Hi Dick,

Thank you so much!

Warm regards,
Bladibla.