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

  • Thread starter VVS
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  • #1
VVS
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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
 

Answers and Replies

  • #2
34,450
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That substitution looks like a good idea, if you split up your integral into two parts first.
 
  • #3
VVS
91
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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.
 
  • #4
34,450
10,566
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.
 

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