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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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Argument of $3+4i$

for $z = \sqrt{3 + 4i}$, I am trying to put this in Standard form, where z is complex. I let $w = 3+4i$ and find that the modulus, $|w|=r$, is 5. I am having trouble solving for arg(w). I find that $\tan^{-1}{\theta} = \frac{4}{3}$. However, this is…
Hlepkit
  • 55
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Simplify a series of $\cos x$ or $\sin x$

Suppose we have this series $$\sum\limits_{k=0}^n a^k \cos(kx) = 1 + a\cos(x) + a^2 \cos(2x) + a^3 \cos(3x) + \cdots + a^n\cos(nx)$$ What should we do to simplify this?
Serius Black
  • 147
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Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$

Let $z_1$ and $z_2$ be 2 elements of $C _∞$ . Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$ (the point on the sphere corresponding to $z_1$ and…
Diane Vanderwaif
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Confused with imaginary calculus

So $i$ is the complex unit and $n \in \mathbb{N} $. $$e^{2 \pi \ n \ i} = 1$$ $$1^{2 \pi \ n \ i} = 1$$ $$(e^{2 \pi \ n \ i})^{2 \pi \ n \ i} = e^{-4\pi^2 \ n^2}$$ $$e^{-4\pi^2 \ n^2} \neq 1$$ I’m confused with this, can someone please explain to my…
solid
  • 220
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Equation in complex numbers

Find all complex numbers such that: $$|z_1|=|z_2|=|z_3|$$ $$z_1+z_2+z_3=1$$ $$z_1\cdot z_2 \cdot z_3=1$$ There is solution with vectors or Vietes formulas. Can we solve this problem with using only algebra?
chaos
  • 599
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How to understand $Re(z) + Im(z) = 2$ & $Re(z) - Im(z) = 0$, $z \in \mathbb{C}$

So I got stuck on a question asking me to plot in $\mathbb{C}$ the following two expressions ($z \in \mathbb{C}$): $$Re(z) + Im(z) = 2$$ $$Re(z) - Im(z) = 0$$ Now this should be easy enough... If we set $z = x +iy$ the first equation should…
Lukas Arvidsson
  • 2,376
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Complex number - prove an inequality

Question: Given that:$$z^n\tan\theta_0 + z^{n-1}\tan\theta_1 + z^{n-2}\tan\theta_2 + ... + \tan\theta_n = 3$$ And that $\theta_i \in (0, \frac{\pi}{4})$, prove that: $$|z| > \frac{2}{3}$$ Approach: I tried to solve the question using $|z_1 +…
Gummy bears
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Complex numbers - minimum value proof

Question: For:$$|z - z_1|^2+|z - z_2|^2+|z - z_3|^2+\cdots+|z - z_n|^2 = S$$ Prove that the minimum value of $S$ is when:$$z = \frac{z_1+z_2+z_3+\cdots+z_n}{n}$$ I have no idea how to even start this question. I tried to do it graphically but…
Gummy bears
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if $\alpha$ $\beta$ and $\gamma$ are the roots of a equation than find the value of .

if $\alpha$, $\beta$ and $\gamma$ are the roots of equation $x^3-3x^2+3x+7=0$ ($\omega$ is the cube root of unity),then $\frac{{\alpha}-1}{{\beta}-1}$+$\frac{{\beta}-1}{{\gamma}-1}$+$\frac{{\gamma}-1}{{\alpha}-1}$ is (a)$\frac{3}{{\omega}}$ (b)…
gaufler
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Help with Complex Number problem (Argand Diagram)

Can anyone help me with this question? "Use De Moivre's theorem to solve the equation $z^5 = 1.$ Show that the points representing the five roots of this equation on an Argand diagram form the ventics of a regular pentagon of Perimeter…
Justin HT
  • 109
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What is this representation of complex numbers?

I was doing numericals on synchronous generators and came across this step in one of the examples. I have no idea what kind of math is used here. Can someone help? $(1.5 + 2.0j)\Omega = 2.5 \angle 53.13^\circ \Omega$ $(0.3+1.22j)\Omega = 1.256…
Ishaan Sejwal
  • 123
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Quadratic formula for $z^2 + (\alpha + \beta i)z + \gamma + i\delta = 0$ where $z\in\mathbb{C}$

The problem statement is to solve the quadratic equation $$ z^2 + (\alpha + i\beta)z + \gamma + i\delta = 0. $$ I took the standard…
dustin
  • 8,241
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Find all solutions for $z^3 = \overline{z}$

I know that $z = a + ib$ and that $\overline{z} = a - ib$, but when I try and calculate the solutions I get an unsolvable equation. Would appreciate any help.
David
  • 73
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4 answers

Compute $z^3 = -26 -18i$

$z^3 = -26 -18i, i =$ imaginary unit. How do I solve this? So far I've calculated the length of it; $|z| = \sqrt{(-26)^2+(-18)^2}=10\sqrt10$ I think I'm supposed to use the polar form of it, but I'm not sure how.
peroxy
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If $u(z) = \frac{az+b}{cz+d}$ is bijective on the upper half complex plane, show that $a,b,c,d$ are real.

The following question is from Greene-Krantz, Function Theory of One Complex Variable (Q 1.10) Let $U = \{ z \in \mathbb{C} \colon \textrm{Im} \, z > 0 \}$. Prove that if $ u(z) =\displaystyle \frac{az+b}{cz+d}$ with $a,b,c,d \in \mathbb{C}$ and $u…
Kelvin Soh
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