Articles of complex analysis

Prove the following equality: $\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi$

This question already has an answer here: Evaluate $\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt$ 4 answers

Complex chain rule for complex valued functions

Let $f=f(z)$ and $g=g(w)$ be two complex valued functions which are differentiable in the real sense, $h(z)=g(f(z))$. Prove the complex chain rule. All partial derivatives: $$ \frac{\partial h}{\partial z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial z} + \frac{\partial g}{\partial \bar w}\frac{\partial \bar f}{\partial z} $$ and $$ \frac{\partial h}{\partial \bar z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial […]

Express $u(x,y)+v(x,y)i$ in the form of $f(z)$

I need to express $f(z)$ from the form $\color{blue}{u+vi}$ to the form $\color{blue}z$ for example if: $g(z)=\frac{1}{x+yi}$ so $ =g(z)=\frac{1}{z}$ $$f(z)=\underbrace {x\sqrt{x^2+y^2}-2x}_{=u(x,y)}+\underbrace {\bigg(y\sqrt{x^2+y^2}-2y+1\bigg)}_{=v(x,y)}i$$ My try: $$x\underbrace{\sqrt{x^2+y^2}}_{=|z|}-2x+y\underbrace{\sqrt{x^2+y^2}}_{=|z|}i-2yi+i$$ $$x|z|-2x+yi|z|-2yi+i$$ I’m stuck here

Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on $X$ that take values in $[0,1]$. Suppose that $\mathscr A\subseteq BC(X)$ is an algebra (that is, it is a vector space that also contains […]

problems with singularity $0$ of $\int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz$.

I have the complex integral \begin{equation*} \int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz \end{equation*} where $W$ is a circle with radius $6$ and centered at $0$. Obviously we have two singularities, one in $0$ and one in $3$. I only get confused by the singularity in $0$. I don’t clearly see what $\lim_{z\rightarrow0}$ of $zf(z)$ is. the limit $ze^\frac{1}{z}$ […]

Explicit formula of Mobius Transformation that maps non intersecting circles to concentric circles

How to find a Mobius Transformation that maps the circles $C(0,1)$ and $C(4,2)$ to a pair of concentric circles. I have already discussed this problem in general case in this blog. By the composition of transformation we can do this. But , the idea I have seems burdensome to calculate. Is there any good way […]

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, $\delta$ is the small radius. We consider $\displaystyle f(z) = \frac{\log^2(z)}{z^2 + 1}$ where $z = x+ iy$ How can we prove: $$\oint_{\Gamma} f(z) dz \to 0 \space \text{when} […]

Analytic function on an open disc.

Let $\mathbb{D}=\{ z\in\mathbb{C}:|z|<1\}$. Which of the following are correct? There exist a holomorphic function $f:\mathbb{D}\rightarrow\mathbb{D}$ with $f(0)=0$ and $f'(0)=2$ There exist a holomorphic function $f:\mathbb{D}\rightarrow\mathbb{D}$ with $f(3/4)=3/4$ and $f'(2/3)=3/4$ There exist a holomorphic function $f:\mathbb{D}\rightarrow\mathbb{D}$ with $f(3/4)=-3/4$ and $f'(3/4)=-3/4$ There exist a holomorphic function $f:\mathbb{D}\rightarrow\mathbb{D}$ with $f(1/2)=-1/2$ and $f'(1/4)=1.$ Option one is not true by […]

Conformal mapping $z+\frac{1}{z}$, how to see the mapping to hyperbolas? p.2. The conformal map $z+\frac{1}{z}$ maps circles $|z|=r$ to ellipses and $arg(z)=\theta$ to hyperbolas. I believe one can display both using the same equations, but I have only managed to display the ellipse and cannot understand the hyperbola. Basically some sources (and I) claim that $$\left(r+\frac{1}{r}\right)\cos\theta+i\left(r-\frac{1}{r}\right)\sin \theta$$ is an equation of an ellipse, when […]

Line passing through origin in $\mathbb{C}^2$ where all numbres $\{z_i\}$ belong to same component of $\mathbb{C} \setminus \ell$?

Let $z_1, \dots, z_n \in \mathbb{C}$ be such that$${1\over{z_1}} + \dots + {1\over{z_n}} = 0.$$Is there a line $\ell$ passing through the origin of the complex plane $\mathbb{C}$ such that all the numbers $z_1, \dots, z_n$ belong to the same component (open half-plane) of $\mathbb{C} \setminus \ell$?