How Can the Series Expansion of Cosec^2 Pi x Be Derived?

[alle.fabbri]

Hi all!
I found on a book of QFT in curved spacetime (Birrel and Davies, pag 53) the following identity
$$cosec^2 \pi x = \frac{1}{sin^2 \pi x} = \pi^{-2} \sum_{k=-\infty}^{+\infty} \frac{1}{(x-k)^2}$$
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

 

[jasonRF]

This can easily be done by contour integration. Consider the function

$$f(z) = \frac{\pi \cot \pi z}{(z-x)^2}$$

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

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.

 

[alle.fabbri]

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 $$k$$ on the real axis I get for them
$$Res[f(z),k]=\underset{z\rightarrow k}{lim} \frac{\pi(z-k)}{tg(\pi z)} \frac{1}{(z-x)^2} = \frac{1}{(x-k)^2}$$
The function has a double pole in $$z=x$$ so there
$$Res[f(z),x]=\underset{z\rightarrow x}{lim} \, \frac{d}{dz} \left[ (z-x)^2 \frac{\pi}{tg(\pi z)} \frac{1}{(z-x)^2} \right] = \underset{z\rightarrow x}{lim} \, \frac{d}{dz} \left[ \frac{\pi}{tg(\pi z)} \right] = -\pi^2 - \frac{\pi^2}{tg^2 \pi x} = -\pi^2 cosec^2 \pi x$$
So one ends up with the desired relation if can prove that the path integral goes to zero as $$N \rightarrow \infty$$. 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 $$R \rightarrow \infty$$??
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...

 

[jasonRF]

Why do you pick such a contour? This machinery could work even if I pick a circle of radius R and then let $$R \rightarrow \infty$$??...

I picked that contour because that is the "standard" contour that I was taught for this. Why does it make sense? First, $$N+1/2$$ is used so that no singularities are on the contour. Second, it isn't too bad so find a constant $$C$$ (independent of $$N$$) such that
$$\sup |\cot \pi z| \leq C$$
for $$z$$ 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

 

[jasonRF]

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.