# Integral of 1/sqrt(x)exp(-ix) dx using integration by parts

1. Mar 28, 2013

### VVS

Hi,

1. The problem statement, all variables and given/known data
I have already evaluated the integral 1/sqrt(x)exp(-ix) using The Residue Theorem and now I was looking for another method. So I thought of applying integration by parts and I got this attached formula.

Now I am wondering how to evaluate this series. My first doubt is whether it converges or not.
Actually I just want to evaluate it for n=1.
Does anyone have any ideas? Can this sum be expressed as an integral?

Thank you

2. Mar 28, 2013

### haruspex

I don't see much hope with that series, but how about applying the 2D polar coordinates trick used to calculate the integral of exp(-x2) from 0 to ∞?

3. Mar 29, 2013

### VVS

Hey,

Yeah I know it is way to complicated.
But one question: If I know the answer of the integral doesn't that mean that that series should converge to that answer?

4. Mar 29, 2013

### haruspex

Assuming no algebraic errors, yes. But it could be that the easiest way to sum the series is by reversing the steps to the integral and solving that.