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 the power of complex number defined yet?

Let $z$, and $b$ be two complex numbers. What is $$f_b(z)=z^b.$$ If I write it like this: $$ \left(re^{i\theta}\right)^{b}=r^{b}e^{ib\theta}. $$ Would this even make sense? Wolframalpha gives me $(-i)^i=e^{\pi/2}$ using the formula above. How to…
zighalo
  • 759
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Proving by calculation that $\arg(-2) = \pi$

The fact that it is true, seems very obvious, if one draws the complex number $z = (-2 + 0i)$ on the complex plane. The angle is certainly 180 degrees, or pi radians. But how can this be proven by calculation? Using $\arg(z)=\arctan(b/a)$ or even…
Alec
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Prove that $z_1, z_2, z_3, z_4$ are the vertices of a rectangle if and only if...

I have to prove that $z_1, z_2, z_3, z_4$, where $|z_1| = |z_2| = |z_3| = |z_4| = 1$, are the vertices of a rectangle if and only if $z_1z_2z_3+z_1z_2z_4+z_1z_3z_4+z_2z_3z_4=0$ Any help? There is a hint in my book about the converse, that says that…
lea
  • 171
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Summation of complex numbers and simplification

By considering $$ \sum_{k=0}^{n-1}(1+i\tanθ)^k\tag{1}$$ Show that $$ \sum_{k=0}^{n-1}\cos(kθ)\sec^kθ=\cotθ\sin(nθ)\sec^nθ\tag{2}$$ Provided $θ$ is not an integer multiple of $\frac{π}{2}$. My take on this was by taking the real part of $(1)$ and…
George
  • 2,556
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absolute value of sum of complex numbers squared

is this correct $ \left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-2|a||b|+|b|^2$ Thank you
Boby
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The principle argument of the product of two complex numbers in the second quadrant

I would like some help to prove the following: Show that, if Re $z_1<0$, Im $z_1>0$, Re $z_2<0$ and Im $z_2 >0$, then Arg$(z_1z_2)=$Arg$(z_1)+$Arg$(z_2)-2\pi$. Thanks for any help in advance.
Jim Darson
  • 177
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Tricky Argand diagram sketch

I'm trying to sketch $D=\{z \in \mathbb{C}: 2 \leq \vert z \vert < \vert z-2 \vert <4\}$. I know that, geometrically, this is all $z$ whose distance from the origin is $\geq 2$ and is $<$ whose distance from $2$, which is less than $4$. The question…
beep-boop
  • 11,595
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prove the following equation about inverse of tan in logarithmic for

$$\arctan(z)=\frac1{2i}\log\left(\frac{1+iz}{1-iz}\right)$$ i have tried but my answer doesn't matches to the equation .the componendo dividendo property might have been used. where $$\arcsin(x)=\frac1i\log\left(iz+\sqrt{1-z^2}\right)$$
gaa
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Exponential of complex variable

What is the equivalent to " $(e i)^z$ " , where i is the imaginary "i" and z is a variable (maybe a complex one) ? (I'm thinking in a possible symmetry with $e^{iz}$)
Luis
  • 41
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Real and Imaginary

$$Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 2$$ $$Im\Big(({\frac{1+i}{1-i})^5\Big)} = 1$$ I got that $Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 1 \ne 2$ And, that $\Big(({\frac{1+i}{1-i})^5\Big)} = i $ , which means that…
Ilan Aizelman WS
  • 4,128
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To find the value of complex number

If $z$ and $w$ are two non zero complex numbers such that $|zw| =1$ and $\arg z - \arg w = \pi/2$ then conjugate of $(zw)$ =?
user142778
  • 61
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Basic complex variable proposition

I have to prove the following property, but no idea how to start... I was told to solve it with polar coordinates, but I still don't know how. Let $\tau$ be a complex number with positive imaginary part. Prove that there exists a $\delta >0$…
Marc
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Calculating absolute value and argument of a complex number

I want to calculate the absolute value and argument of the complex number $a = \left(\sqrt{3} - i\right)^{-2}$. In order to calculate these two values I tried to reform the number into the form $z = x + y \cdot i$: $$a =\left(\sqrt{3} -…
muffel
  • 2,879
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Finding argument of a complex number

How do you evaluate the following $$\text{Arg}\{\sin\frac{8\pi}{5} + i(1 + \cos\frac{8\pi}{5})\}$$
user34304
  • 2,749
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Complex No.s Proving Question

This is a problem from G.Stephenson's Mathematical Methods for Science Students which I am stuck on: Prove that, if $z = \cos \theta + i \sin \theta$ and $ n $ is any positive integer, $ z^n - \frac {1}{z^n} = 2i \sin n \theta$ Show that $$ z^{2n}…
user141554
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