Articles of complex analysis

What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I fail to see the intuition behind this. Most of the time, I even think they tend to make calculations harder. Is there a simple interpretation […]

Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it’s called the Riemann hypothesis, which asserts: $$ \zeta(s) = 0 \Rightarrow \operatorname{Re} s = \frac1 2 \text{ or } \operatorname{Im} s = 0 .$$ Now there are numerical methods for approximating $\zeta(s)$, but as I understand, […]

Can $ \int_0^{\pi/2} \ln ( \sin(x)) \; dx$ be evaluated with “complex method”?

Can the following integral be evaluated using complex method by substituting $\sin(x) = {e^{ix}-e^{-ix} \over 2i}$? $$ I=\int_0^{\pi/2} \ln ( \sin(x)) \; dx = – {\pi \ln(2) \over 2}$$

$f^3 + g^3=1$ for two meromorphic functions

Can you find two non-constant meromorphic functions $f,g$ such that $f^3 +g^3=1$?

Sum of nth roots of unity

Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+…+c^{n-1} = 0$ Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}{n}}}{1-e^{\frac{i2\pi}{n}}} =0$$ by use of the sum of geometric series’. My issue is proceeding further to show that this is indeed true.

How to rigorously justify “picking up half a residue”?

Often in contour integrals, we integrate around a singularity by putting a small semicircular indent $\theta \rightarrow z_0 + re^{i\theta}$, $0 \leq \theta \leq \pi$ around the singularity at $z_0$. Then one claims that the integral “picks up half a residue” as $r \rightarrow 0$, so we compute the residue, divide by two, and multiply […]

Non-existence of a bijective analytic function between annulus and punctured disk

Suppose $A=\{z\in \mathbb{C}: 0<|z|<1\}$ and $B=\{z\in \mathbb{C}: 2<|z|<3\}$. Show that there is no one -to-one analytic function from A to B. Any hints? Thanks!

contour integral with singularity on the contour

I want to compute the following integral $$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$ The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie on the curve $|z|=1$ so I can’t apply residue formula. How can I proceed?

Conformal maps from the upper half-plane to the unit disc has the form

Prove that the conformal maps from the upper half-plane $\mathbb{H}$ to the unit disc $\mathbb{D}$ has the form $$e^{i\theta}\dfrac{z-\beta}{z-\overline{\beta}},\quad\theta \in \mathbb{R} \text { and }\beta \in \mathbb{H}.$$ Any hints?

Laurent series for $1/(e^z-1)$

Trying to compute the first five coefficients of the Laurent series for $$\frac{1}{e^z-1}$$ centered at the point $0$. I’m not seeing a way to use the geometric series due to the exponential. Any ideas?