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

In summary, The conversation discusses the evaluation of a series using different methods and the question of whether or not it converges. One person suggests using the 2D polar coordinates trick, while the other points out that the answer to the integral should also be the answer to the series.
  • #1
VVS
91
0
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.

integral.jpg


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
 
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  • #2
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
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
VVS said:
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.
 

1. What is the formula for integrating 1/sqrt(x)exp(-ix) dx using integration by parts?

The formula for integrating 1/sqrt(x)exp(-ix) dx using integration by parts is ∫ u dv = uv - ∫ v du, where u = 1/sqrt(x) and dv = exp(-ix) dx.

2. What is the first step in using integration by parts to solve this integral?

The first step is to choose u and dv in a way that will simplify the integral. In this case, u = 1/sqrt(x) and dv = exp(-ix) dx would be a suitable choice.

3. How do you determine the value of du and v when using integration by parts?

To determine the value of du, take the derivative of u with respect to x. In this case, du = -1/2x^(3/2) dx. To determine the value of v, integrate dv with respect to x. In this case, v = -iexp(-ix).

4. What are the limits of integration when using integration by parts?

The limits of integration remain the same throughout the process. In this case, the limits would be from 0 to ∞.

5. Can integration by parts be used to solve more complex integrals?

Yes, integration by parts can be used to solve more complex integrals by choosing u and dv in a way that simplifies the integral as much as possible. It is a useful tool in finding antiderivatives and in solving definite integrals.

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