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

[VVS]

Hi,

Homework Statement

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

 

[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 ∞?

 

[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?

 

[haruspex]

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?

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.