Integral of 1/sqrt(x)exp(-ix) dx

[VVS]

Hi,

I am just doing this out of curiousity.

Homework Statement

I want to integrate 1/sqrt(x)exp(ix) dx from minus infinity to infinity.

Homework Equations

The Attempt at a Solution

I had a couple of ideas one was to substitute x=u^2
but then you mess up the limits and you get minus imaginary infinity.

The other idea was to use Jordan's Lemma. But as far as I know 1/sqrt(x) doesn't have a residue so it can't be applied.

How do you solve this integral then?

thank you

 

[mfb]

That substitution looks like a good idea, if you split up your integral into two parts first.

 

[VVS]

Hey,

Thanks for the hint. I have done the splitting, but I am not sure about the limits.

If I set u=sqrt(x) then the lower limit is plus or minus imaginary infinity and moreover the upper limit can take two values: plus infinity or minus infinity.

I don't know how to integrate that. I am aware that cos(x^2)dx and sin(x^2)dx are the well known fresnel integrals.

thanks for your help.

 

[mfb]

For x from 0 to plus (real) infinity, u goes from 0 to plus (real) infinity.
For negative x, you can transform the integral to the integral for positive x.