Articles of complex analysis

Complex – Entire functions

How can I prove this. I could not use $\Im(w)<0$ condition in Liouville’s theorem. Let $f(z)$ be an entire function and assuming that $f(z)$ does not take values in $\Im(w)<0$ show that $f$ is identically zero. Thanks.

Residue at essential singularity

I need a little help with the following problem. I’ve tried many ways, but i didnt succeed. I think there needs to be a trick or something, some transformation. The task is to find the residue of the function at its singularity e.g. z=-3 \begin{equation} f(z)=\cos\left(\frac{z^2+4z-1}{z+3}\right) \end{equation} I tried to write it as \begin{align} \cos\left(\frac{z^2+4z-1}{z+3}\right)=1-\frac{1}{2!}\left((z+1)-\frac{4}{z+3}\right)^2+\frac{1}{4!}\left((z+1)-\frac{4}{z+3}\right)^4-\frac{1}{6!}\left((z+1)-\frac{4}{z+3}\right)^6+\ldots […]

How to show that $f$ is constant by using Liouville's theorem?

If $f$ is entire and $\mbox{Arg}(f(z))=-\frac{\pi}{2}$,when $|z|=1$ then show that $f$ is constant. All I need to prove is that f is bounded but I can’t figure out how.I’d like someone’s help.

Formula for the series $f(x):=\sum\limits_{n=1}^\infty\displaystyle\frac{x}{x^2+n^2}$

Good evening! Recently I had to study the properties of a function $f(x):=\sum\limits_{n=1}^\infty\displaystyle\frac{x}{x^2+n^2}$. I found its supremum and proved that $\lim\limits_{x\to+\infty}f(x)=\displaystyle\frac{\pi}{2}$. But as far as I remember such sums can be calculated explicitly. Maybe someone knows a quick hint to this? And in general: a lot of sums can be calculated explicitly by reducing to […]

Taylor series expansion of $\log$ about $z=1$ (different branches)

I realize this is not the fastest way of getting a Taylor’s series expansion of $f(z)=\log(z)$ about $z=1$. But here goes. I am assuming I am working on the principal branch of the logarithm ($-\pi<\theta<\pi$). I am assuming that $f(1)=\log(1)=0$. That’s the branch I am on. Next, some derivatives: $f'(z)=1/z$ $f”(z)=-1/z^2$ $f”'(z)=2/z^3$ $f^{(4)}(z)=-6/z^4$ … $f^{(n)}(z)=(-1)^{n+1}(n-1)!/z^n$ […]

differential system on the torus

In a recent topic I’ve studied on complex analysis I had to study the differential system on the torus $\mathbb T^2:$ $$\begin{cases}\frac{\partial}{\partial y}u-\frac{\partial}{\partial x}v=\sin(y)-\cos(x),\\\\ \frac{\partial}{\partial x} u+\frac{\partial}{\partial y}v=0,\end{cases}$$ with the conditions $$\int_0^{2\pi}\int_0^{2\pi}u(x,y)\mathrm d x\mathrm dy=\int_0^{2\pi}\int_0^{2\pi}v(x,y)\mathrm d x\mathrm dy=0.$$ In particular it seemed to me that this system was explicitly solvable and to do so i […]

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be true numerically. Can anyone offer some advice on how to approach proving (or disproving) this? It originates from a question involving the (complex) Hilbert transform of a symmetric non-increasing probability distribution: \begin{equation} h(y) = (1+\text{i}y)\int_{-\infty}^\infty \frac{1}{1 + \text{i}(y-t)}\text{d}G(t) […]

How to properly translate the coefficients of a Taylor series?

Given a Taylor series $$f(z) = \sum_{k=0}^\infty c_k^{(a)}\frac{(z-a)^k}{k!}$$ of a meromorphic function $f$ in $\mathbb C$ (i.e. analytical except for a set of isolated points) around some value $a\in\mathbb C$, where $c_k^{(a)}=\frac{d^k}{da^k}f(a)$, one can naively apply the translation operator $\exp((b-a)\partial_c)|_{c=a}$ on $f(a+z)$ to obtain the series expansion around another point $b$: $$\begin{align} f(b+z) &\ =\ […]

Explicit fractional linear transformations which rotate the Riemann sphere about the $x$-axis

Background: Let $S^2$ denote the unit sphere in $\mathbb{R}^3$. By “stereographic projection”, I mean the mapping from $S^2$ (remove the north pole) to the complex plane which sends \begin{align*} \begin{bmatrix} x \\ y \\ t \end{bmatrix} \in S^2 \mapsto z = \frac{x+iy}{1-t} \in \mathbb{C}. \end{align*} The inverse mapping, from the complex plane to the sphere, […]

Operational details (Implementation) of Kneser's method of fractional iteration of function $\exp(x)$?

For a long time (a couple of years) I’m following the Q&A’s about “half-iterate of $\exp(x)$” etc. where there exists a $\mathbb C \to \mathbb C$ due to Schröder’s method, but also a $\mathbb R \to \mathbb R$ for fractional heights $h$ due to Hellmuth Kneser. I would like to understand the latter method of […]