Why doesn't the residue theorem work for branch cut integration?

In summary, the conversation is about a problem involving branch cut integration with a given equation for 0< \alpha <1. The person used a keyhole contour but ended up with a different result than expected. They were helped by someone who pointed out their mistake in assuming that the lower and upper part of the branch cuts were the same. After realizing their mistake, the answer became trivial.
  • #1
gonzo
277
0
I need help with a branch cut intgration. The problem is to show the following for [itex]0< \alpha <1[/itex]:

[tex]
\int_{0}^{\infty}{x^{\alpha - 1} \over x+1}={\pi \over sin\alpha\pi}
[/tex]

I used the standard keyhole contour around the real axis (taking that as the branch cut), but using the residue theorem I end up with:

[tex]
-\pi i e^{i\alpha\pi}
[/tex]

Which obviously doesn't match. Although this does match up for alpha equals one half.

Some help would be appreciated.
 
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  • #2
It would be easier to see what went wrong if you showed more work!

What was your residue? (I think you were Ok here)

How did you deal with the part of the keyhole contour that lies just below the x-axis? I think this is what went wrong. On this part you will be working with a different branch of the logarithm, so z^{alpha} will take on different values here than the bit above the x-axis.
 
  • #3
Thanks, you actually showed me my stupid mistake. I had just worked through the problem with alpha as one half, in which case the lower and upper part of the branch cuts are the same (well, negatives, but going in different directions, so you get a divisor of 2).

I had naively assumed that it would be the same for this problem. Your comment inspired me to take a closer look and I realized it wasn't the case, and then the answer was trivial.

Thanks!
 

1. What is branch cut integration?

Branch cut integration is a mathematical technique used to evaluate complex integrals that have branch points, or points in the complex plane where the function becomes multi-valued. It involves choosing a branch cut, or a line or curve in the complex plane, and then integrating along that cut to obtain the desired result.

2. Why is branch cut integration necessary?

Branch cut integration is necessary because certain functions, such as logarithmic and trigonometric functions, are multi-valued in the complex plane. This means that they have more than one possible output for a given input, making traditional integration methods ineffective. Branch cut integration allows for the evaluation of these complex integrals by taking into account the branch points of the function.

3. How do you choose a branch cut for integration?

The choice of a branch cut for integration depends on the specific function being integrated. In general, the branch cut should be chosen in such a way as to avoid any singularities or branch points of the function. It should also be chosen to ensure that the function remains single-valued along the cut.

4. What are some common challenges with branch cut integration?

One common challenge with branch cut integration is determining the appropriate branch cut for a given function. This can require a thorough understanding of the function and its complex behavior. Another challenge is ensuring that the function remains single-valued along the chosen branch cut, as this is necessary for the integral to be well-defined.

5. Are there any alternative methods to branch cut integration?

Yes, there are alternative methods to branch cut integration, such as contour integration and residue calculus. These methods are also used to evaluate complex integrals, but they may be more suitable for certain types of functions or integrals than branch cut integration. It is important for a scientist to be familiar with a variety of integration techniques in order to choose the most appropriate method for a given problem.

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