Using Residue Theorem To Prove Value Of Summation

In summary, uploading images with $\LaTeX$ has some disadvantages, and people responding to posts cannot use the quote feature to edit expressions.
  • #1
joypav
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  • #2
Hello!

I just want to point out that uploading images rather than using $\LaTeX$ has some disadvantages. Each user is allowed to upload a certain total number of bytes of attachments (we cannot offer every user unlimited storage space), and so at some point you will run into the ceiling and not be able to upload any more images. Better to be aware of this now rather than to find this out at the point where you cannot upload an image. Also, when people respond to your posts, they cannot use the quote feature to edit your expressions/content...they must instead type everything out themselves.:)
 
  • #3
Hi joypav,

Note that $\cot \pi z$ has singularities at the integers. What is the residue of $\dfrac{\pi \cot \pi z}{z^2}$ at $z = n$ for some integer $n$?
 
  • #4
MarkFL said:
Hello!

I just want to point out that uploading images rather than using $\LaTeX$ has some disadvantages. Each user is allowed to upload a certain total number of bytes of attachments (we cannot offer every user unlimited storage space), and so at some point you will run into the ceiling and not be able to upload any more images. Better to be aware of this now rather than to find this out at the point where you cannot upload an image. Also, when people respond to your posts, they cannot use the quote feature to edit your expressions/content...they must instead type everything out themselves.:)

Yes, I know... I've been wanting to switch over. I use LaTeX some, for school and such, but I just need to look at how to do that here.

I hadn't considered how it would inconvenience you. I am sorry for that! I appreciate all of the help that I have received here.

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Euge said:
Hi joypav,

Note that $\cot \pi z$ has singularities at the integers. What is the residue of $\dfrac{\pi \cot \pi z}{z^2}$ at $z = n$ for some integer $n$?

I got it.. thank you!
 
  • #5
joypav said:
Yes, I know... I've been wanting to switch over. I use LaTeX some, for school and such, but I just need to look at how to do that here.

I hadn't considered how it would inconvenience you. I am sorry for that! I appreciate all of the help that I have received here.

Okay, as long as you're aware of the uploading limit, that was my primary concern...I didn't want you to be "surprised." :) I'm hoping you will find using $\LaTeX$ here is easy, and if you have any questions, please feel free to post here:

http://mathhelpboards.com/math-formulas-mathjax-62/

As for the "inconvenience," that would only pertain to a small number of posts and I didn't mean to make it out as more than it is. Your posting style is excellent otherwise, and normally I would wait until a user has several hundred posts under their belt before "prodding" them to use $\LaTeX$, but I mostly didn't want you to get caught off-guard by the attachment limit. :)
 

FAQ: Using Residue Theorem To Prove Value Of Summation

What is the Residue Theorem?

The Residue Theorem is a mathematical tool used in complex analysis to evaluate certain integrals, particularly those around closed curves. It states that the integral of a function around a closed curve is equal to the sum of the residues of the function at its isolated singularities inside the curve.

How is the Residue Theorem used to prove the value of summation?

The Residue Theorem can be used to evaluate certain infinite sums by converting them into complex integrals. By finding the residues of the function being summed, we can then use the Residue Theorem to evaluate the integral, which in turn gives us the value of the summation.

What are the prerequisites for using the Residue Theorem to prove summation?

In order to use the Residue Theorem to prove the value of summation, one needs to have a solid understanding of complex analysis, including concepts such as complex numbers, complex functions, and contour integration. It is also helpful to have knowledge of Laurent series and poles of functions.

Can the Residue Theorem be used for all types of summation?

No, the Residue Theorem can only be used for certain types of summation, specifically those that can be converted into complex integrals. This means that the series must have a finite number of terms and follow a specific pattern, such as geometric or telescoping series.

Are there any limitations to using the Residue Theorem for proving summation?

Yes, there are limitations to using the Residue Theorem for proving summation. As mentioned before, it can only be used for certain types of series. Additionally, the series must converge in order for the Residue Theorem to be applicable. It also requires a good understanding of complex analysis and can be quite complex and time-consuming to use.

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