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 […]

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

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

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.

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 […]

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!

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?

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?

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?

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Struggling with “technique-based” mathematics, can people relate to this? And what, if anything, can be done about it?