Articles of complex numbers

Complex Numbers on a Circle – Challenging Problem

The following is given: $z^{142}+\frac{1}{z^{142}} (z\neq 0,z\in \mathbb{C})$ A) Prove that for every complex number z on the unit circle, this expression is real. B) Is it possible that the expression is real for every z on a circle with radius unequal 1 ? C) Calculate the expression if z is a root of the […]

What steps are taken to make this complex expression equal this?

How would you show that $$\sum_{n=1}^{\infty}p^n\cos(nx)=\frac{1}{2}\left(\frac{1-p^2}{1-2p\cos(x)+p^2}-1\right)$$ when $p$ is positive, real, and $p<1$?

Proving the Mandelbrot set is bounded

I am trying to prove that the Mandelbrot set defined as the set $\mathcal M$ of complex numbers $c$: the recursive sequence defined as $$z_0=c, \space \space \space z_{n+1}={z_n}^2+c$$ is bounded. Prove that $\mathcal M \subset \{|z|\leq 2\}$. I’ve tried to show this by the absurd: take $c \in \mathcal M$ : $|c|>2$. Now I […]

How does $\cos x=\frac12(e^{ix}+e^{-ix})$?

I have seen the following definition many times: $$\cos x=\frac12(e^{ix}+e^{-ix})$$ However, it makes little sense to me as it appears far from obvious. Please help me understand this definition, either a derivation or explanation of how the values on the right equal $\cos x$.

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ \sum_{p=1}^{d-1}\frac{\cos(\frac{2\pi p}{d})}{\tan^k(\frac{\pi p}{d})} $$ is 0 for every odd k, and if anyone knows, what will the value be at even $k$, since they turn out to be “nice” when I compute […]

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

How is it solved: $\sin(x) + \sin(3x) + \sin(5x) +\dotsb + \sin(2n – 1)x =$

This question already has an answer here: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? 5 answers

Cauchy-Schwarz in complex case, using discriminant

There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v – w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz inequality. In trying to do the same thing in the complex case, I ran into some trouble. First, there are proofs here, […]

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 $\mathbb{C} \setminus \ell$?