Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Is it true that $e^{i\omega}=1$ for any value of $\omega$?

I have a problem understanding the polar form of a complex number. We have that $$e^{i\omega} = \cos{\omega}+i\sin{\omega}.$$ In particular, if $\omega=2\pi$ then $e^{i2\pi} = \cos{2\pi}+i\sin{2\pi}=1$. Yet, if I rewrite it like this: $$e^{i\omega}…
aukxn
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Showing that $\frac{z^n}{1+z^{2n}}$ is real for $|z|=1$.

Can you help me to prove that $$\frac{z^n}{1+z^{2n}}$$ is a real number, given that $z$ is a complex number with modulus $1$ and $n$ a positive integer, such that $z^{2n}$ is not equal to $-1$.
user290335
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If complex no. ($z$) satisfying $\frac{1}{2}\leq |z|\leq 4\;,$ Then Max. and Min. of $\left|z+\frac{1}{z}\right|$

Let $z$ be a complex no. satisfying $\displaystyle \frac{1}{2}\leq |z|\leq 4\;,$ then the Sum of greatest and least value of $\displaystyle \left|z+\frac{1}{z}\right|$ is $\bf{My\; Try::}$ Let $z=re^{i\theta} = r\left(\cos \theta+i\sin…
juantheron
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Generalized addition function

I would like to have an example (or a proof that there does not exist) of a function on the complex numbers, which for lack of a better term I'll call generalized addition, such that $$x\oplus y=y\oplus x$$ $$(x\oplus y)\oplus z=x\oplus(y\oplus…
Ivan
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Solve the binomial equation

Solve the binomial equation $$z^4 = -8$$ Below is the steps i have done 1: I have taken |-8| that is 8 and then done 8^(1/4) which is 2^(1/4). 2: Since $z=r(cos\alpha+isin\alpha)$ leads me to $r^4(cos4\alpha+isin4\alpha)=-8(cos\pi/2+isin\pi/2)$…
addde
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Intuition behind $i^{i}$.

My query is about the $i^{i}$ , where $i$ is defined to be the imaginary unit, and $i \in C$. I know the proof of this value, we just have to substitute $i$ as $\large{e^{i\left(\frac{\pi}{2}+2n\pi\right)}}$. Where $n$ is any integer because all…
Someone
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The image in $\mathbb{C}$ of $\mathbb{R}^2$ under a map of counterpropagating plane waves is...?

Define $$f_n(\mathbf{r})=\frac{1}{n}\sum_{k=1}^n\exp\left(2\pi i\binom{\cos\left(2\pi k/n\right)}{\sin\left(2\pi k/n\right)}\cdot\mathbf{r} \right)$$ as the sum of $n$ counterpropagating plane waves. Then, let $I_n^R(z)\geq0$ be the "density of…
DumpsterDoofus
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A small complex number whose total distance from other given complex numbers is large

Let $z_1,z_2,...,z_n$ be distinct complex numbers such that $|z_i|\leq1$. Is it true that there exists $z, |z|\leq1$ such that $\displaystyle\sum_{i=1}^n |z-z_i|\geq n$ ? Thank you.
Bless
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Find all solutions to the following equation: $x^3=-8i$

Find all solutions to the following equation: $$x^3=-8i$$ I found the modulus, $$r=8$$ $$\operatorname{arg}(x)=\arctan(-8/0)=-π/2+2πk$$ By De Moivre's Theorem: $$2[\cos(-π/6+2/3πk)+i\sin(-π/6+2/3πk)]$$ First solution I got is: $$\sqrt3-i$$Is my…
Mohamed
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Equation with complex numbers

Solve the following equation in $ \mathbb{C} $: $ |z - |z + 1|| = |z + |z - 1|| $ I started it but I don't know how to finish it. Here is what I did so far: $ |z - |z + 1||^2 = (z - |z + 1|)(\bar{z} - \overline{|z + 1|}) = |z|^2 - z \cdot…
George R.
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How to show that $\sqrt[3]{-1+\sqrt{-7}}+\sqrt[3]{-1-\sqrt{-7}}$ is a real number at a time before the invention of complex numbers

I have read this PDF from ocw.mit.edu about complex numbers. There is one interesting question: Imagine yourself at the time, when complex numbers had to be invented yet. How to show that $$\sqrt[3]{-1+\sqrt{-7}}+\sqrt[3]{-1-\sqrt{-7}}$$ is a real…
user50224
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Considering $ (1+i)^n - (1 - i)^n $, Complex Analysis

I have been working on problems from Complex Analysis by Ahlfors, and I got stuck in the following problem: Evaluate: $$ (1 + i)^n - (1-i)^n $$ I have just "reduced" to: $$ (1 + i)^n - (1-i)^n = \sum_{k=0} ^n i^k(1 - (-1)^k) $$ by using expansion…
David Cardozo
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Algebraic complex numbers $z$ satisfying the equations $z^n+\bar{z}^n = z+\bar{z}$ for all positive $n$

Is there a complex number $z$ such that $z\neq 0,1$ and $z^n +\bar{z}^n = z+\bar{z}$ for all positive integers $n$? Is there an algebraic complex number $z$ such that the above properties hold? The existence of such a complex number would amaze me.
Ali
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Find a value for a number to the power of a complex number

Find a value for $2^{-4i}$? I have no idea what to do or how to find the value. My thoughts are that I should use logarithm. Can someone please show me how to solve this?
jimi
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If $|z +\frac{1}{z}|=a$ find extreme values ​​of $|z|$

Let $a> 0$. Knowing that $z$ is complex number with $|z +\frac{1}{z}|=a$ find extreme values ​​of $|z|$. My partial solution: $$|z +\frac{1}{z}|=a \iff (z +\frac{1}{z})(\overline{z} +\frac{1}{\overline{z}})=a^2\iff |z|^2+\frac{1}{|z|^2} +…
medicu
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