Articles of complex analysis

Determining holomorphicity

I need to determine where the following function is differentiable and holomorphic in $\mathbb C$: $$f(z)=(z-3)^i$$ I have the derivative as $df/dz= i(z-3)^{-1+i}$. The answer in my book says f is differentiable and holomorphic on $\mathbb C$ where $y\neq0$ and $x>3$. I don’t see where this comes from. wolframalpha plotted the derivative and I can […]

Motivation for Mobius Transformation

Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the motivation to study Mobius Transformation? Why should one look at the map defined in the above way?

Lagrange inversion theorem application

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show me the step by step process by which $$\sum_{i>-1}b_iu^i = t$$ is obtained. I can seem to find any links […]

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\nabla u)=(\gamma u_x)_x+(\gamma u_y)_y=0,\\ u(x+2\pi,y)=\mu u(x,y),\\ u(x,1)=0,\\ u(x,0)=0 \mbox{ or } u_y(x,0)=0. \end{cases}$$ Let’s denote the set of possible values of the eigenvalue parameter $\mu$, for which the system has a solution $\mu_\gamma$. Does there exist a conductivity $\alpha(y)$ on […]

Integrating $\int_\Gamma1/z \, dz$ Through a “Branch Cut”

In my complex analysis book, there is an example where I am asked to compute $\int_\Gamma1/z \, dz$ for two cases: in both of them, $\Gamma$ is a curve going from $-i$ to $i$ in the complex plane. However, in the first case, $\Gamma$ lies in both the first and fourth quadrants, crossing the positive […]

Understanding a Proof for Why $\ell^2$ is Complete

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I’m trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I’m trying to show that $\ell^2$ is complete in a particular way outlined below. I only used the first few steps of the proof because once I understand the third […]

Mapping circles using Möbius transformations.

I need some help with the following problem from Ahlfors’ Complex Analysis. Problem: Find a single Möbius transformation $\phi$ (that is, a map of the form $\phi(z) = \dfrac{az + b}{cz + d}$, where $a,b,c,d$ are complex numbers) that maps the circles $|z| = 1$ and $\left|z – \frac{1}{4}\right| = \frac{1}{4}$ to concentric circles. Infinite […]

Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1(t)=t$ for $0\leq t \leq R$, $\gamma_2(t)=Re^{i\frac{2\pi}{3}t}$ for $0\leq t \leq 1$, and $\gamma_3(t)=(1-t)Re^{i\frac{2\pi}{3}}$ for $0\leq t \leq 1$. So, the contour is a wedge, and by letting $R\rightarrow \infty$ we’re integrating over one […]

analytic in open unit disk,corresponding to a bounded sequence and a bounded functional sequence

Let $f$ be analytic in open unit disk, we need to show there exist $\{z_n\}$ with $|z_n|<1$ and $|z_n|\rightarrow 1$ then $f(z_n)$ is bounded. could any one give me Hints for this one?

Banach-Space-Valued Analytic Functions

This is Chapter VII, $\S$3, exercise 4, from Conway’s book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic if the limit $$\lim_{h \to 0} \frac{ f(z+h)-f(z)}{h}$$ exist in $X$ for all $z \in G$. Prove that if […]