Articles of complex analysis

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 […]

Fast calculation for $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$

I want to show that $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$, but is there a faster method than finding the contour and doing all computations? Otherwise my idea is to do the substitution $x=e^t$, integral than changes to $\int _{-\infty}^{\infty}\frac{t e^t}{1+e^{2t}}dt$. Next step is to take the contour $-r,r,r+i\pi,-r+i\pi$ and integrate over it…

Non Existence of a proper holomorphic map from the punctured unit disc to an Annulus

Show that there is no proper holomorphic map from the punctured unit disc to an annulus $A_r=\{z \in \mathbb C:1 <|z| < r \}$. Def:A map $f: X \to Y$ is called proper if $f^{-1}(K)$ is compact for every compact set $K$ in Y. please give some hints/ideas to prove this.Can someone please give a […]

Prove $e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$

It seems to me that $$e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$$ This integral seems to converge for all $x\in\mathbb{C}$ I came upon this conjecture by following the instructions here to do a half integral twice. Can anyone prove this conjecture is true?

A strange identity related to the imaginary part of the Lambert-W function

Working on a problem in QFT, i was stumbeling about some expressions containing the Lambert-$W$ function. Playing around, i discovered experimentally that the following statement seems to be true $$ \Im (W_0(-x))=-\Im (W_{-1}(-x))\,\,\, \text{if} \,\,\, x>1/e $$ For large $x$ we can write $W_0(-x)\approx\log(-x)$ and $W_{-1}(-x)\approx\log(-x)-2 \pi i$ choosing now $\log(-x)=\pi i+\log(x)$ gives the desired […]

Use Picard's Theorem to prove infinite zeros for $\exp(z)+Q(z)$

Suppose that $Q(z)$ is a nonconstant polynomial. Then show that the function $$f(z)=\exp(z)+Q(z)$$ has infinitely zeros. My idea is to show that $\infty$ is an essential singularity thus by Picard’s theorem $f(z)$ assumes every complex number infinitely times except on possible value. I was stuck with the possibility that $0$ may be the exception, if […]

Residue of $f(z) = \frac{z}{1-\cos(z)}$ at $z=0$

I’ve been self-studying some complex variables and just started on residues. I’m looking at a problem where I’ve been asked to calculate the residue of: $$f(z) = \frac{z}{1-\cos(z)}$$ at $z=0$. I’m not really sure how to find the Laurent series of this function, and I can’t really apply Cauchy’s integral formula either. So I would […]

Isolated singularities of the resolvent

Let $T$ be a bounded operator on $l_2$ such that there exists $\mu$ in the spectrum of $T$ which is an isolated point of the spectrum. We know that for any $x\in l_2$ the resolvent function $R(x):\rho(T)\to l_2$ given by $R(x)(\lambda)=(\lambda I-T)^{-1} x$ is analytic, and so $\mu$ is an isolated singularity of $R(x)$. Is […]