Chapter10 - Chapter 10 Discrete Applications of the Residue...

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Chapter 10 Discrete Applications of the Residue Theorem All means (even continuous) sanctify the discrete end. Doron Zeilberger On the surface, this chapter is just a collection of exercises. They are more involved than any of the ones we’ve given so far at the end of each chapter, which is one reason why we lead the reader through each of the following ones step by step. On the other hand, these sections should really be thought of as a continuation of the lecture notes, just in a different format. All of the following ‘problems’ are of a discrete mathematical nature, and we invite the reader to solve them using continuous methods—namely, complex integration. It might be that there is no other result which so intimately combines discrete and continuous mathematics as does the Residue Theorem 9.9. 10.1 Infinite Sums In this exercise, we evaluate—as an example—the sums k 1 1 k 2 and k 1 ( - 1) k k 2 . We hope the idea how to compute such sums in general will become clear. 1. Consider the function f ( z ) = π cot( πz ) z 2 . Compute the residues at all the singularities of f . 2. Let N be a positive integer and γ N be the rectangular curve from N +1 / 2 - iN to N +1 / 2+ iN to - N - 1 / 2 + iN to - N - 1 / 2 - iN back to N + 1 / 2 - iN . (a) Show that for all z γ N , | cot( πz ) | < 2. (Use Exercise 31 in Chapter 3.) (b) Show that lim N →∞ R γ N f = 0. 3. Use the Residue Theorem 9.9 to arrive at an identity for k Z \{ 0 } 1 k 2 . 4. Evaluate k 1 1 k 2 . 5. Repeat the exercise with the function f ( z ) = π z 2 sin( πz ) to arrive at an evaluation of X k 1 ( - 1) k k 2 . 104
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CHAPTER 10. DISCRETE APPLICATIONS OF THE RESIDUE THEOREM 105 ( Hint : To bound this function, you may use the fact that 1 / sin 2 z = 1 + cot 2 z .) 6. Evaluate k 1 1 k 4 and k 1 ( - 1) k k 4 .
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