Articles of complex analysis

Let $f(z)$ be a one-to-one entire function, Show that $f(z)=az+b$.

Let $f(z)$ be a one-to-one entire function, Show that $f(z)=az+b$. My try : Because $f$ is entire it has a taylor series around zero (in particular). $f(z)=\sum^{\infty}_{k=0} a_kz^k$ Proof by contradiction : let $m \geq 2$ Suppose $a_m \neq 0 $ and $ f(c)=f(b) \Rightarrow \ \ \ 0= f(c)-f(b)= \sum^{\infty}_{k=0} a_k(c^k-b^k ) \therefore \ […]

Inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$ (Bromwich integral)

I am looking for the inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$. I started by using the general formula of the Bromwich integral: $\displaystyle \lim_{R\to\infty} \int_{a-iR}^{a+iR} \frac{1}{2\pi i}\log\left(\frac{s+1}{s}\right) e^{st}ds $ Then, I used that: $\displaystyle \log\left(\frac{s+1}{s}\right)=\sum_{n=1}^{\infty} \frac{ (-1)^{n+1} }{n} (1/s)^n $ for $|s|>1$. Since $|s|>1$ the Bromwich line should be to the right of $1$. So: $\displaystyle […]

Complex Analysis: Liouville's theorem Proof

I’m being asked to find an alternate proof for the one commonly given for Liouville’s Theorem in complex analysis by evaluating the following given an entire function $f$, and two distinct, arbitrary complex numbers $a$ and $b$: $$\lim_{R\to\infty}\oint_{|z|=R} {f(z)\over(z-a)(z-b)} dz $$ What I’ve done so far is I’ve tried to apply the cauchy integral formula, […]

Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$

I can’t do this exercise of Conway’s Book: For $r>0$ let $A=\{w:w=\exp(1/z), 0<|z|<r\}$, determine the set $A$. Any hints?

Analytic Capacity

For a compact set $K\subset\mathbb{C}$ the analytic capacity is defined as $$\gamma(K)=\sup\{|f^\prime(\infty)|:f\in M_K\}$$ where $M_K$ is the set of bounded holomorphic functions on $\mathbb{C}\backslash K$ with $\|f\|_\infty\le 1$ and $f(\infty)=0$. I have two questions. What is the intuition behind this definition? What does $f^\prime(\infty)$ mean? It doesn’t seem to be $\lim_{z\rightarrow\infty}f^\prime(z)$ so I’m confused. It […]

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor’s Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How do we know that Taylor’s series even hold for complex numbers? How can we justify the substitution of a complex number into the series?

Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$

Evaluate by complex methods $$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$ Sis.

Question Relating with Open Mapping Theorem for Analytic Functions

This problem is taken from Section VIII.4 of Theodore Gamelin’s Complex Analysis: Let $f(z)$ be an analytic function on the open unit disk $\mathbb{D}=\{|z|<1\}$. Suppose there is an annulus $U = \{r<|z|<1\}$ such that the restriction of $f(z)$ to $U$ is one-to-one. Show that $f(z)$ is one-to-one on $\mathbb{D}$. Any hints?

Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) \overline{\varphi_k(\zeta)},\qquad(z,\zeta\in\Omega)$$ where $\{\varphi_k\}_{k=1}^\infty$ is any orthonormal basis of the Bergman space $A^2(\Omega)$ of Lebesgue square-integrable holomorphic functions on $\Omega$. This series representation converges at least pointwise, since the […]

Finding closed forms for $\sum n z^{n}$ and $\sum n^{2} z^{n}$

Using the identity $\frac{1}{1-z} = 1 + z + z^2 + \ldots$ for $|z| < 1$, find closed forms for the sums $\sum n z^n$ and $\sum n^2 z^n$. My solution: Because $\displaystyle1 + z + z^{2} + \ldots = \frac{1}{1-z}$, $\displaystyle1 + 2z + 3z^2 + \ldots = \sum_{n=1}^\infty n z^{n-1} = \frac{1}{(1-z)^2}$ and […]