Serie convergent to cosec

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In summary, the conversation discusses a mathematical identity found in a book about quantum field theory in curved space. The identity is derived using contour integration and the residue theorem, and the integral is shown to go to zero as the size of the contour approaches infinity. The conversation also mentions some recommended books for complex analysis, including "Introduction to Complex Analysis" by Priestley, "Complex Variables" by Fisher, and "Complex Variables with Applications" by Ablowitz and Fokas.
  • #1
alle.fabbri
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Hi all!
I found on a book of QFT in curved spacetime (Birrel and Davies, pag 53) the following identity
[tex]
cosec^2 \pi x = \frac{1}{sin^2 \pi x} = \pi^{-2} \sum_{k=-\infty}^{+\infty} \frac{1}{(x-k)^2}
[/tex]
Can anyone help to derive it or give some reference to a book for the proof. I have no idea of how prove this...
Thanks
 
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  • #2
This can easily be done by contour integration. Consider the function

[tex] f(z) = \frac{\pi \cot \pi z}{(z-x)^2} [/tex]

Integrate it around the square contour defined by the corners [tex](\pm 1 \pm i) (N+1/2) [/tex], use the residue theorem, and take the limit as [tex] N \rightarrow \infty [/tex].

You will have to prove that the integral goes to zero in the limit, which is the only "tricky" part.

If you haven't seen this before, most complex analysis books cover summation of series by contour methods.
 
Last edited:
  • #3
Thank you for the answer!
I think I got the idea underlying your advice. Let me work it out.
Since the simple pole of the function are the integers [tex]k[/tex] on the real axis I get for them
[tex]
Res[f(z),k]=\underset{z\rightarrow k}{lim} \frac{\pi(z-k)}{tg(\pi z)} \frac{1}{(z-x)^2} = \frac{1}{(x-k)^2}
[/tex]
The function has a double pole in [tex]z=x[/tex] so there
[tex]
Res[f(z),x]=\underset{z\rightarrow x}{lim} \, \frac{d}{dz} \left[ (z-x)^2 \frac{\pi}{tg(\pi z)} \frac{1}{(z-x)^2} \right] = \underset{z\rightarrow x}{lim} \, \frac{d}{dz} \left[ \frac{\pi}{tg(\pi z)} \right] = -\pi^2 - \frac{\pi^2}{tg^2 \pi x} = -\pi^2 cosec^2 \pi x[/tex]
So one ends up with the desired relation if can prove that the path integral goes to zero as [tex]N \rightarrow \infty[/tex]. I have only a question left. Why do you pick such a contour? This machinery could work even if I pick a circle of radius R and then let [tex]R \rightarrow \infty[/tex]??
Since I studied my complex analysis exam on the notes given by my professor, I never looked for such books...can you address me giving some authors that you think are the best for this topic?
Thanks again...
 
  • #4
alle.fabbri said:
Why do you pick such a contour? This machinery could work even if I pick a circle of radius R and then let [tex]R \rightarrow \infty[/tex]??...
I picked that contour because that is the "standard" contour that I was taught for this. Why does it make sense? First, [tex]N+1/2[/tex] is used so that no singularities are on the contour. Second, it isn't too bad so find a constant [tex]C[/tex] (independent of [tex]N[/tex]) such that
[tex]\sup |\cot \pi z| \leq C [/tex]
for [tex]z[/tex] on the countour. Proving that a circular contour goes to zero in the limit is probably more work than for the square contour.

Since I studied my complex analysis exam on the notes given by my professor, I never looked for such books...can you address me giving some authors that you think are the best for this topic?
Thanks again...

There are so many reasonable books on complex analysis, and everyone likes different styles. For the summation "trick" specifically, almost all books have it, but I don't recall any books having more than one or two pages on this. Some books relegate it to the exercises. So don't buy a book just for this trick, only buy a book if you want a reference or a fun read. Note that for alternating series, you can use the cosecant instead of the cotangent.

Regarding specific books, I always go to "introduction to complex analysis" by Priestley first. Not because it is so good (it is fine but nothing special), but because it was the main textbook when I took the class so after 100+ hours with it I can easily pick it up and understand it. Fisher's Complex Variable book (cheap Dover) is quite good, but is not the best for multiple valued functions. Dettman's cheap "applied complex variables" is pretty complete, but fairly dry. My favorite intro books are probably Saff and Snyder (sp?) and the book by Ablowitz and Fokas. Used copies of old editions is the way I always go whenever possible, as it can save a bundle of money. Schaum's outline is okay, too.

good luck
 
  • #5
A few more books ...

Churchill and Brown's "complex variables" book is a standard. I have the fifth edition and it is reasonable. I just looked - in the 5th edition series summation is only in a couple of problems.

Carrier, Krook and Pearson is full of super challenging problems (no solutions). The presentation of the theory in that book is so quick that most of us could never learn the theory there. I really like their contour integration, integral transform techniques, and conformal mapping chapters. Of course, the best thing about the chapters are the examples and the challenging problems.

Whittaker and Watson's "course of modern analysis" book is a classic that includes hundreds of pages of exposition on the special functions that physicists see a lot. It is also full of very hard problems. But this is a great place to look for derivations related to Bessel, Hypergeometric, Elliptic, Theta, and Gamma functions, as well as many more. The first edition of this book came out over a hundred years ago, but it is still fun.
 

1. What is a "serie convergent to cosec"?

A "serie convergent to cosec" refers to a mathematical series that converges to the cosecant function. This means that as the number of terms in the series increases, the value of the series approaches the value of the cosecant function at a given point.

2. How do you determine if a series is convergent to cosec?

To determine if a series is convergent to cosec, you can use the ratio test or the comparison test. The ratio test involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, then the series is convergent to cosec. The comparison test involves comparing the given series to a known convergent series. If the known series converges, then the given series also converges to the same function.

3. What are some real-world applications of series convergent to cosec?

Series convergent to cosec have applications in various fields such as physics, engineering, and economics. For example, in physics, they are used to calculate the electric field generated by a charged ring. In engineering, they are used to calculate the deflection of a beam under a distributed load. In economics, they are used to model the growth of a population or the value of a financial investment over time.

4. Can a series convergent to cosec diverge?

Yes, a series convergent to cosec can diverge if the series does not satisfy the necessary conditions for convergence, such as the terms not approaching zero or the series being an alternating series with non-decreasing terms.

5. Are there any other functions that can be represented by a series?

Yes, there are many other functions that can be represented by a series, such as sine, cosine, logarithmic, and exponential functions. These series are important in mathematics and have various applications in different fields.

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