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 […]

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 $p<1$?

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 […]

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 $\cos x$.

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 […]

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 […]

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

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

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, […]

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

Intereting Posts

How do I tell if matrices are similar?
Prove combinatorics identity
How to solve this recurrence Relation – Varying Coefficient
How can I evaluate $\lim_{x\to1}\frac{\sqrt{5-x}-2}{\sqrt{2-x}-1}$ without invoking l'Hôpital's rule?
Simplify the surd expression.
Rudin Series ratio and root test.
The Green’s function of the boundary value problem
Trigonometric/polynomial equations and the algebraic nature of trig functions
Fourier cosine transform
What is Octave Equivalence?
Combinatorics: How to find the number of sets of numbers in increasing order?
Using variation of parameters, how can we assume that nether $y_1$, $y_2$ equal zero?
Definition of General Associativity for binary operations
For what values of $x$ is $\cos x$ transcendental?
probability that he will be selected in one of the firms