# Tagged Questions

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### What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$\int_{0}^\infty dx\, R(x),$$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
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I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ... 1answer 150 views ### Proofs that there is no f(z) such that \exp f(z) = z for all z \in \Bbb{C}\setminus\{0\} When I first learned about this result I was completely stunned that there is no holomorphic function f(z) on \Bbb{C}\setminus\{0\} such that \exp f(z) = z. What are some interesting proofs of ... 8answers 520 views ### Final year project ideas - complex analysis For my final year, I have to do a project for a module. I want to investigate something in the complex analysis area. I've only covered the basics of analysis, like Cauchy's IT/IF, residue theorem ... 6answers 12k views ### “Where” exactly are complex numbers used “in the real world”? I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ... 6answers 2k views ### Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ... 4answers 447 views ### Counterexamples in complex analysis In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex ... 1answer 60 views ### Re[f(x + n i)] = 0 and f(z) is not periodic Let z be a complex number and f(z) an entire function such that For x real and n any integer. Re[f(x + n i)] = 0 and f(z) is not periodic. What are typical examples of such f(z) ? Is ... 3answers 233 views ### Applications of complex variables beyond undergrad syllabus So complex numbers solve all polynomials, appear as eigenvalues, appear in intermediate calculations in solving cubics, relate trig to hyperbolic functions, can be used to contour integrate real ... 1answer 746 views ### Expository articles on Analysis and Probability theory When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ... 5answers 704 views ### Primer on complex analysis and Riemann surfaces for undergraduate physics / theoretical physics majors Ref: The Road to Reality: a complete guide to the laws of the universe, (Vintage, 2005) by Roger Penrose [Chap. 7: Complex-number calculus and Chap. 8: Riemann surfaces and complex mappings] I'm ... 4answers 463 views ### Detecting a negative coefficient in a power series Suppose that I have an analytic function f(z)=\sum_{n=0}^\infty a_n z^n which converges on some disk around the origin. For a particular function I encountered, I wished to prove that every ... 5answers 917 views ### Proving the identity \sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}. Can you help prove the functional equation:$$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$Specifically, I am looking for a solution using complex ... 21answers 19k views ### Different methods to compute \sum\limits_{n=1}^\infty \frac{1}{n^2} As I have heard people did not trust Euler when he first discovered the formula$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}. However, Euler was Euler and he gave other proofs. I ...
Suppose that a rational generating function is given representing the function $f(x) = \sum_{i=0}^{2n}{c_i x^i}$ where $c_i \in \mathbb{N}$ The goal is to determine if the coefficient in the middle, ...