Articles of complex analysis

Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$

Show that $\int^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$ by integrating $z^{-1}e^{iz}$ around a closed contour $\Gamma$ consisting of two portions of the real axis, from -$R$ to -$\epsilon$ and from $\epsilon$ to $R$ (with $R > \epsilon > 0$) and two connecting semi-circular arcs in the upper half-plane, of respective radii $\epsilon$ and $R$. […]

Form of most general transformation of the upper half plane to the unit disk.

In David Blair’s book on Inversion Theory, he write that the transformation $$ T(z)=e^{i\theta}\frac{z-z_0}{z-\bar{z}_0} $$ is the most general transformation mapping the upper half plane to the unit circle, provided $z_0$ is in the upper half plane. If $z_0$ is in the lower half plane, then the upper half plane is mapped to the exterior […]

Iterative roots of sine

Is there an analytical function $f(z)$ such that $f(f(z)) = \sin(z)$? More generally, an analytical function such that f applied $n$ times to $z$ gives $\sin(z)$? Is there a general theory for answering this question for functions besides $\sin(z)$?

If $\operatorname{Re}f^\prime > 0$ on a convex domain, then $f$ is one-to-one.

Let $f(z)$ be analytic on a convex region $D \subset \mathbb{C}$. If $\mathrm{Re}f'(z)>0,\forall z\in D$, then show that $f(z)$ is a one-to-one function, that is, if $z_1\ne z_2,$ then $f(z_1)\ne f(z_2)$.

Hard integral that standard CAS get totally wrong

How to solve the following integral: $$\int_{-\infty }^{\infty }\exp \left ( i\left ( ax^3+bx^2 \right ) \right )dx$$ Standard CAS seem to get it totally wrong, see: So what is the right ansatz and solution? EDIT There seems to be a problem with the way this question is posed… which I quite frankly don’t […]

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 […]