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.

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Inequality in the complex space

Can someone please help me to prove the following inequality: let $a,b$ be any two complex numbers and $p>1$ we have $\left||a|^{p-1}a-|b|^{p-1} b\right|\leq c\left(|a|^{p-1} +|b|^{p-1}\right)|a-b|$ where $c$ is a constant that depends on…
user317150
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real life application of imaginary numbers

I am having a hard time understanding what is an imaginary number, and the purpose for studying them. I can't stop my mind from thinking "if it doesn't exist and is not 'real' why is it being studied"? Can someone please give an example of a…
devakotia
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How to compute with roots of negative numbers

I have heard that taking roots of negative numbers gives me multiple values, like $\sqrt{-1}=\pm i$. But how can I compute expressions that takes multiple times of numbers having several values? Like is $\sqrt{-1}+\sqrt{-1}=\pm i+\pm i=-2i,0,2i$? I…
selflearning
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Real part of $cis (\theta)$

I have a question on a proof regarding this: The question is "Show this is true," although I am not sure if the last minus sign should be an equal sign, or else it doesn't make sense. I know that this is related to De Moivre's theorem and Euler's…
CCC
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Describe the set of all complex numbers $z$ such that $|z-a |+| z-b |=c$

Describe the set of all complex numbers $z$ such that : $$|z-a |+| z-b |=c$$ where $a,b, c$ are real At a simple look I immediately recognized that this is some ellipse because it's the same with the definition of the ellipse with two foci,…
Admir Kullolli
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Show that $|z^2| + Re(a \cdot z) +b = 0$ has only solutions for $|a^2|\geq 4b$

The Question: show that $|z^2| + Re(a \cdot z) +b = 0$ only has a solution for $|a^2|\geq 4b$ $a \in \Bbb{C}$ $b \in \Bbb{R}$ Question 1.10 If I substitute $z=r_z e^{i\phi_z}$ and $a=r_a e^{i \phi_a}$ I get $r_z^2 + Re(r_z r_a…
oldmansaur
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$\arg(z_1+z_2)=0$ and $\text{Im}(z_1z_2)=0$. Prove that $z_1=\bar{z_2}$

$\arg(z_1+z_2)=0$ and $\text{Im}(z_1z_2)=0$. Prove that $z_1=\bar{z_2}$ I substituted $z_1=a+ib$ and $z_2=c+id$ Using first condition, $$z_1+z_2=\bar{z_1}+\bar{z_2}$$ $$a+ib+c+id=a-ib+c-id$$ $$b+d=0$$ So $z_1=a+ib$, $z_2=c-ib$ Using second…
Aditya Dev
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What is the difference in notation for Ln(z) vs. ln(z)

We are learning about the principal vs. non-principal values of the natural logarithm. There are 2 problems: $\ln(-7)$ $\operatorname{Ln}(-7)$ I solved $\ln(-7)$ for: $$=\ln|-7|+i\pi+2\pi k$$ Is the difference in notation for the other problem…
whatwhatwhat
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Convert $e^z$ to Cartesian form (complex numbers)

Convert $e^z$ to $a+bi$ I'm having trouble figuring out this very simple problem. Below is my attempt, but can you really have $1/e$ as the modulus of a complex…
whatwhatwhat
  • 1,587
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How do I express this $f(x,y)$ in terms of $f(z)$

$$f(x,y)=e^y\sin x+ie^y\cos x$$ The problem requires me to express in terms of $z$ only. My attempt: $$=e^y(\sin x+i\cos x)$$ $$=ie^y(\cos x-i\sin x)$$ If $e^{-i\theta}=(\cos\theta - i\sin \theta)$, then: $$=ie^ye^{-i\theta}$$ But then I get stuck.…
whatwhatwhat
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$(1+i)(e^{(1+i)\phi})$ expressed in polar and rectangular form

$$(1+i)(e^{(1+i)\phi})$$ I need to express this in both polar and rectangular form, but the difficult part is that extra $i$ above the $e$. Also, what am I supposed to make of $\phi$? We normally use $\theta$ to represent the argument of a complex…
whatwhatwhat
  • 1,587
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Limit of $\exp(z^2)$ as $|z|$ tends to infinity

Let $g(z) = \exp(z^2)$ and $L$ a ray starting at the origin. Determine those $L$ along which $g$ has a limit (finite or infinite) as $|z|$ tends to infinity and $z ∈ L$. Find the value of the limit also. Please I need help.
UserAb
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Solve for $z$ (complex numbers)

$$ z^3=i $$ The problem simply states to solve for $z$, but I know that there is some concept to be practiced here about the nth roots of unity. I'm just beginning to learn this concept so I didn't really know how to go about solving this, but below…
whatwhatwhat
  • 1,587
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Complex number - wrong result at the end

I need to solve this: $$ \frac{i^4+3}{i-1}$$ On my book the result should be: $-2-2i$ but I get: $-1-2i$ and I do not understand where the error is. My steps: $$ \frac{i^4+3}{i-1} = \frac{i^4+3}{i-1} \cdot \frac{-1-i}{-1-i}$$ $$ \frac{(i^4+3)(-1-i)…
Christian Giupponi
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If $z = r[\cosθ +i\sinθ]$, show that $w = \sqrt[n]{r}[\cos(θ/n) +i\sin(θ/n)]$

a. If $z=r\left[\cos\theta +i\sin\theta \right]$, then $ w =\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}\right) +i\sin\left(\frac{\theta }{n}\right)\right] $ is an $n$-th root of $z$, where $r\geq 0$ Now my first week of abstract algebra just…
Charlene
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