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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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To prove that a polynomial is divisible by another polynomial .

$x^{4p}$+$x^{(4q+1)}+x^{(4r+2)}+x^{(4s+ 3)} $is divisible by x^3+x^2+x+1, where ,q ,r ,s belongs to Natural numbers. So , I did is this : $x^3+x^2+x+1$ = $(x^2+1)(x+1)$ , So , x = +1 and -1. Then , put in f(1) and f(-1). But I am not able to solve…
S.M.T
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Guidance for a complex number proof

I am given $\left(x+iy\right)^{\frac{1}{3}}=a+ib$, and I need to prove $4\left(a^{2\ }-b^{2}\right)=\frac{x}{a}+\frac{y}{b}$. The first "key" thing (I hope it's actually useful!) I notice is that we are only considering the real parts (I'm pretty…
user71207
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Image of a region under a Mobius transformation

I have a function $$f(z) = \frac{z+1}{i(z-1)}.$$ And I have a triangle enclosed by the vertices $0$, $1$ and $i$. I am trying to determine the image of the triangle under $f$ on the standard complex plane. I also know that $f$ is a mobius…
user773827
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find the roots $(2z+3)^3=\frac{1}{64}$

There are 3 roots 1 real and 2 imaginary i found one z by doing $\frac{\frac{1}{4}-3}{2}$ so $z=\frac{-11}{8}$ however there are two more complex roots which are $z=\frac{-25+i√3}{16}$ and $z=\frac{-25-i√3}{16}$ but i dont know how to get to it…
Sara
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Show that $\sin6\alpha\equiv \sin2\alpha(16\cos^4\alpha-16\cos^2\alpha+3)$

$$\sin6\alpha\equiv \sin2\alpha(16\cos^4\alpha-16\cos^2\alpha+3)$$ Can you help me with De Moivre's theorem and how I would go about tackling this question. I understand that De Moivre's theorem states that $(\cos\alpha+i\sin\alpha)^n \equiv \cos…
maxmitch
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Fixed points on complex function $T(z) = (1+i)z + 3-4i$

If $T(z) = (1+i)z + 3-4i$ is a complex function, and I got this fixed point: $z=4+3i$ Is this correct? Are there more fixed points? $(1+i)z+3-4i=z$ $3-4i=z-(1+i)z$ $3-4i=-iz$ $z=-\frac{3-4i}{i}=4+3i$
user899924
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How do you divide a complex number with an exponent term?

Ok, so basically I have this: $$ \frac{3+4i}{5e^{-3i}} $$ So basically, I converted the numerator into polar form and then converted it to exponent form using Euler's formula, but I can have two possible solutions. I can have $5e^{0.972i}$ (radian)…
Gladstone Asder
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Representation of a complex number in the Argand Plane

Consider the complex number $\omega = \frac{1+i}{\sqrt{2}} $. Find the value of $(\sum_{k=0}^n z^{k^2}) (\sum_{j=0}^n z^{-j^2})$ I started by collecting the terms with similar but opposite sign indices and writing the sum in polar form, but I am…
Ashwin Krishnan
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Real analysis when solving a complex number problem?

We are to solve $(x-1)^3+8=0$ in the complex set to do so we bring 8 to the RHS and cube root on both sides $$(x-1)=2(-1)^{1/3}$$ The RHS is of three different forms $(-1,\omega,\omega^2)$ But the LHS is just written in a single form why do we do…
Glowingbluejuicebox
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Proof of $ \arg (G(j \omega))=-\omega\left(t_{2}-t_{1}\right) $

We have this formula: $$ \displaystyle \arg (G(j \omega))=-\omega\left(t_{2}-t_{1}\right) $$ (We understand the amplitude of course, please help us to find a proof for the phase shift) without a proof (control theory course). Can someone remind us…
Dovendyr
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Simplifying $z^2+i=0$

I need to simplify $z^2+i=0$ and find all solutions for $z$. I have seen that the solutions to $z=\sqrt{i}=\left(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\right)$ and $\left(-\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}\right)$. I was hoping to find a…
Obsessive Integer
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If two complex numbers are close enough then their Arguments too?

Let $z_1=r_1e^{i\theta_1}$ and $z_1=r_2e^{i\theta_2}$ be two complex numbers such that $-\pi<\theta_1,\,\theta_2\leq \pi$. My question is: If $|z_1-z_2|<\delta$, for some $\delta>0$ sufficiently small then $|\theta_1-\theta_2|<\lambda$ for some…
ElliptCg
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For a function $f(x)$, does $f(ix)$ have any graphical meaning?

For a function $f(x)$, if I graph $f(2x)$, then it would look like $f(x)$ but compressed by a factor of $2$. If I graph $f(\frac{x}{2})$, then it would be stretched. I am very curious to know whether we can say something similar graphically about…
Akil
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How to find all third roots of $-i$?

How to find all third roots of $-i$? I have no idea. Could you give me some hint?
hide on bush
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$a=\frac{1+i}{\sqrt2}$ $b=\frac{\sqrt3+i}{2}$ $a,b,z\in\mathbb{C}$ What is the real part of $z=\frac{a-b}{1+ab}$?

$a=\frac{1+i}{\sqrt2}$ $b=\frac{\sqrt3+i}{2}$ $a,b,z\in\mathbb{C}$ What is the real part of $z=\frac{a-b}{1+ab}$ ? The answer is $0$ but i do not know why. I tried simply substituting $a, b$ but i didn't get anything in a simple form. Then i tried…
Pete42
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