0
votes
0answers
31 views

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 ...
8
votes
1answer
436 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

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) ...
2
votes
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 ...
6
votes
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 ...
10
votes
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 ...
25
votes
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 ...
11
votes
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 ...
0
votes
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 ...
4
votes
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 ...
21
votes
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 ...
3
votes
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 ...
11
votes
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 ...
14
votes
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 ...
224
votes
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 ...
4
votes
2answers
152 views

What is an alternative formulation to a contour integral?

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, ...
9
votes
5answers
474 views

Visualising functions from complex numbers to complex numbers

I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for ...