# Questions tagged [complex-numbers]

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

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### Trying to solve $|2x-15| = -x^2 - 5x -8$

My first instinct was to take the positive and negative of the right hand side, resulting in $2x-15 = -x^2 - 5x - 8$, and $2x-15 = x^2 + 5x + 8$, which results in the first giving me two real answers ...
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### If $a^2 + a + 1 = 0$ find $a^3$

$$a^2 + a + 1 = 0$$ $$(a^2 + a+1) (a-1) = 0(a-1)$$ $$a^3 - 1 = 0$$ $$a^3 = 1$$ This is how I had solved the question by using the identity :- $$a^3 - b^3 = (a - b)(a^2 + b^2 + ab)$$ But the roots of ...
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### Continuity of the complex arcsin function

Palka writes in An Introduction to Complex Function Theory (pp. 84) that Because the principal square root function has its only discontinuities at the points of the negative real axis, it is easy ...
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### Show that no non-zero integers satisfy this pair of equations (from Baltic Way 2021)

Show that no non-zero integers $a, b, x, y$ satisfy: \begin{cases} ax-by=16. \\ ay+bx=1. \end{cases} From Baltic-Way 2021. \begin{align} &(a+bi)(x+yi)=(ax-by)+i(ay+bx)=16+i. \\ &|(a+bi)(x+yi)|...
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### Proving inequality using root of unity

Let $\omega$ be a complex cube root of unity. It can be shown that if $a,b,c \in \mathbb{R}$, then $$(a+b+c)(a+b\omega+c\omega^2)(a+b \omega^2 + c \omega)= a^3+b^3+c^3-3abc$$ I was wondering if this ...
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### For which $s\in\mathbb{C}$ do we have $\sum_{n=0}^\infty{s \choose n}=2^s$?

By comparison with the binomial expansion of $(1+x)^s$, a sufficient condition for the formula $\sum_{n=0}^\infty{s\choose n}=2^s$ to hold is that $A_s=\sum_{n=0}^\infty{s\choose n}$ is absolutely ...
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### Let u and v be the real and imaginary component of f defined by the function. [closed]

$f(z) = \begin{cases}\frac{\dot{z}^2}{z}, & z \ne 0 \\ 0, & z=0 \end{cases}$ where as $\dot{z}$ is the conjugate of $z$. Is the Cauchy Riemann equation satisfied at $(0,0)$? Is $f$ ...
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### Real part of $\quad 1- \frac{2}{\pi} \arctan(r^{\rho}e^{i\rho\theta}).$

To solve the Dirichlet problem using mellin transform, i needed to find the real part of $\quad 1- \displaystyle\frac{2}{\pi} \arctan(r^{\rho}e^{i\rho\theta}).$ I already know the result will be \...
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### $\theta$ is real if and only if $e^{i\theta}$ is in circle

Let us define $e^z = \displaystyle \sum_{k=0}^{\infty} \frac{z^n}{n!},$ where $z$ is a complex number. I want to show that, $\theta$ is real if and only if $e^{i\theta}$ is in circle. That is, I want ...
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### What is this relationship between trigonometric and hyperbolic function?

In the following, I don't understand how they put $\tan{\phi} = \sinh{\frac{\psi}{\sqrt{2}}}$. Is there a relationship?
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### Find area of the region in complex plane $\{z+z^2/2: |z| < 1\}$ [closed]

What is the area of the following region in $\mathbb{C}$? $$\{ z + \tfrac{1}{2}z^2: |z| \leq 1 \}$$
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### Question about specific equality of complex functions under these conditions

Let $f$ and $g$ be two complex functions such that they are both holomorphic in $|z|<4$ and equal in $|z|<1$. The question is: i) Is $f(3i)=g(3i)$ necessarily true? ii) Is $f(3+3i)=g(3+3i)$ ...
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### What is the meaning of roots for $p(x)$ derived from the extension field $F[x]/p(x)$?

In my Analysis class, a section demonstrates how the extension field $\mathbb{R}[x]/(x^2+1)$ provides a root for $x^2 + 1$. In an example leading up to this formal definition for the complex numbers, ...
Let $f(z)=\frac{1}{z}$. Let $\Omega$ be the domain of complex numbers such that $\mathrm{Re}(z)>2$. It's clear that $f$ is continuous on $\Omega$. We can also write $f$ as: $$f(z)=\frac{1}{z}=\frac{... • 417 0 votes 0 answers 19 views ### Determining the image of S = \{z\in\mathbb{C}\mid -3\leq\mathrm{Re(z)} \leq 3, 0\leq \mathrm{Im(z)} \leq 1\} under f(z) = z^2 Let S = \{z\in\mathbb{C}\mid -3\leq\mathrm{Re(z)} \leq 3, 0\leq \mathrm{Im(z)} \leq 1\} and define f(z) = z^2. I am trying to determine description of the image f[S] using a suggested method in ... • 311 0 votes 2 answers 43 views ### The Sum of the n nth Roots of Unity (concerns the definition of the principal root) I was trying to answer the following question from Adam's Calculus and have a problem with the hint it provides. I have read the similar threads and my question does not concern the procedure of ... 1 vote 3 answers 57 views ### What is the real and imaginary part of (i+1)^{17}? [duplicate] Hello i know that (i+1)^{17}=256+256i is. But how do you get to that result? -2 votes 0 answers 40 views ### Is there ever a situation where a \Re(f(x) z)≠f(x)\Re(z) Is there ever a case where if you have a complex function f(z) that can be broken up into a real function times a complex function where you can't just say that taking the real part of it isn't ... • 117 0 votes 4 answers 99 views ### If x^{3}+1=0 ; x \neq -1, then find the value of \sum_{i=1}^{2020} x^{i} I tried solving this question by considering the roots of unity, but somehow I got -(1455+x) as the answer. How can i solve this by polynomial reduction or similar techniques? 0 votes 0 answers 78 views ### Mathematical formulae for Complex Arcus Cotangents? I'm writing reusable complex functions for use in Shaders. I've done most of the complex series. All sines, cosines, and tangents (arcus, hyperbolic, and hyperbolic arcus variations as well) However, ... -1 votes 2 answers 59 views ### Is this set non-existent or simply excluding two points? The set is simply defined as: |z-a|=|z-b| where z,a,b\in \mathbb{C} and a\neq b. Now for z=a, we have: |a-a|=0=|a-b| \rightarrow a=b. This contradicts our hypothesis. The same happens for z=... • 417 4 votes 1 answer 88 views ### (complex)Let \{a_{jk}\}_{j,k \in Z_+} be a double sequence of complex numbers. Prove that... Let \{a_{jk}\}_{j,k \in Z_+} be a double sequence of complex numbers. Prove that if \sum_{j=0}^{\infty}\sum_{k=0}^{\infty} \vert a_{jk}\vert \lt \infty, then \sum_{j=0}^{ \infty}\sum_{k=0}^{\... • 153 1 vote 0 answers 52 views ### A geometric question on complex numbers Suppose z_1,z_2,\cdots,z_n be n complex numbers and$$ r=\max | z_i-z_j|,\;i,j=1,2,\dots,n\;i \neq j.$$Further let$$z=\frac{z_1+z_2+\cdots+ z_n}{n}.$$Is it true that for all k=1,2,..,n,$$ |z-...
Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ find all possible ...