Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
17,696
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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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6
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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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If all roots of equation $z^4+az^3+bz^2+cz+d=0\;(a,b,c,d \in \mathbb R)$ are of unit modulus, then which of the following is/are correct
If all roots of equation $z^4+az^3+bz^2+cz+d=0\;(a,b,c,d \in \mathbb R)$ are of unit modulus, then
(A) $|a|\leq4$
(B) $|b|\leq4$
(c) $|b|\leq6$
(d) $|c|\leq4$
Solution given in book:
$z_1+z_2+z_3+z_4=-...
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Prove $\left|\frac{a-b}{1-\bar{a}b}\right| <1$ if $|a|<1$ and $|b|<1$
Prove $\left|\frac{a-b}{1-\bar{a}b}\right| <1$ if $|a|<1$ and $|b|<1$
a,b are complex numbers.
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If $D \subset \mathbb C\setminus\{0\}$ is an open connection....
a) If $D \subset \mathbb C\setminus\{0\}$ is an open connection. Show that if $\theta_1$ and $\theta_2$ are argument branches in $D$, then there is $k \in Z$ such that $\theta_1(z) = \theta_2(z) + 2k\...
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1
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Taking a simple derivative
I have the following function that I wish to take the derivative of
$$
z(\omega)=\frac{1-C_{c}L_{r}\omega^{2}-\frac{\omega^{2}}{\omega_{r}^{2}}}{i\omega C_{c}\left(1-\frac{\omega^{2}}{\omega_{r}^{2}}\...
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Prove that $|z_{1}+z_{2}| < |1+\overline{z_{1}}\cdot z_{2}|\;$ if $|z_{1}| < 1$ and $|z_{2}| < 1$.
Given $z_{1},z_{2} \in \mathbb{C}$
Prove that $|z_{1}+z_{2}| < |1+\overline{z_{1}}\cdot z_{2}|\;$, if $|z_{1}| < 1$ and $|z_{2}| < 1$.
My Professor gave my classmates and I, the following ...
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$\cosh z = 1$, $\sinh z = 1$ [closed]
University preparation question they gave me.
Find the complex roots of $\cosh z = 1$ and $\sinh z = 1$.
For $\cosh z = 1$, I ended up with $(e^z-1)^2 = 0$, so $e^z = 1$ and hence $z = 0$.
For $\sinh ...
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3
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Convert a pure imaginary number to polar form
I know that when converting a complex number of the form $ z = a + bi $, you need to take the $\arctan$ of $\frac{b}{a}$. But what do you do if $a = 0$?
I know that the pure imaginary numbers live on ...
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Find all polynomials $p(x) \in \mathbb{C}[x]$ such that $p(\mathbb{R}) \subset \mathbb R $ and $p(\mathbb{C - R}) \subset \mathbb{C - R }$.
Find all polynomials $p(x) \in \mathbb{C}[x]$ such that $p(\mathbb{R}) \subset \mathbb R $ and $p(\mathbb{C - R}) \subset \mathbb{C - R }$.
Note that $ \mathbb C =\{a+bi\mid a,b \in\mathbb R \}$.
For ...
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Moebius transformations preserving unit circle
Find all Moebius Transformations preserving unit circle
Note: I am more interested if I got these computations right than the answer.
Approach-1
From page-124 of Needham, a general moebius ...
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(complex)Let $f: D \subset \mathbb R^2 \to \mathbb R^2$ be a function defined on an open subset $D \subset \mathbb R^2$...
Let $f: D \subset \mathbb R^2 \to \mathbb R^2$ be a function defined on an open subset $D \subset \mathbb R^2$. Remember that $f$ is said to be differentiable (in the Frechét sense) at (x_0, y_0) \in ...
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Calculate the length of hypotenuse of a right triangle in the complex plane
As the Pythagorean theorem does not work,
Base = 1
Altitude = i
Hypoteneuse^2 = 1^2 + i^2 = 0?
How can this be calculated?
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Show that $\lim_{z \to 0 }\frac{\log(z+1)}{z} = 1$ for the complex logarithm
Show that $\lim_{z \to 1 }\frac{\log(z)}{z} = 1$ for the complex logarithm,
with the definition of the complex logarithm being
$$ \log(z) = \log |z| + i \arg(z). $$
Edit. Ok, so it seems that it was ...
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Geometry in complex numbers.
Let $\theta_1, \theta_2, \theta_3, … , \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+ \theta_3+… + \theta_{10}=2\pi$. Define complex numbers $z_1=e^{i\theta_1}$, $z_k=...
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Proving that $\arg(w)\in\left(\frac{-3\pi}{4},-\frac{\pi}{2}\right)$
I'm having some trouble with the following exercise:
Let $z_1=-3+2i$, $z_2=-1+2i$ and $z_3=2-i$. Let $$w=\frac{z_1\cdot z_2}{z_3}$$
Show that the following is true without using a calculator: $$|w|=\...
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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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49
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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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I cannot figure out why I get differing results for $\int _\gamma z\,dz$, $\gamma (t)=te^{it}$ when using integration by parts and primitive of $z$
Let $\gamma(t) =te^{it}$, $t\in [0,\pi ]$ and consider $\displaystyle \int \limits _\gamma z\,dz$. Since the primitive of $z$ is $\frac{z^2}{2}$, we can evaluate $\displaystyle \int \limits _\gamma z\,...
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Steepest descent for Linearized KdV equation
I take steepest descent on Linearzied KdV equation,
$$
u_t+u_{xxx}=0
$$
And by Fourier transform I know the phase is
$$
i(k^3+k\frac{x}{t})
$$
I want to know asymptotic of the exponential integral
$$
\...
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Define the domain in which $f(z)=z\cdot \text{Im} (z)$ is differentiable and calculate its derivative.
Define the domain in which the below function is differentiable and calculate its derivative:
$$f(z)=z\cdot \text{Im} (z)$$
I tried checking the analyticity of the function by definining $z=x+iy$, I ...
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2
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Show continuity of the following functions: a) $f: \mathbb C \to \mathbb C, f(z) = \vert z \vert$. b) $g: \mathbb C \to \mathbb C, g(z) = \ \bar z$
Show continuity of the following functions:
a) $f: \mathbb C \to \mathbb C, f(z) = \vert z \vert$.
b) $g: \mathbb C \to \mathbb C, g(z) = \ \bar z$.
a) A function $f$ is continuous if $\lim_{z \to ...
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Square root of a complex number with infinitesimal imaginary part
I am going through lecture in Quantum Field Theory and I am encountering the square root of a complex number with infinitesimally small imaginary part: $$\lim_{y\rightarrow0^+}\sqrt{-x+iy}=-i\sqrt{x}$$...
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if we let $\sqrt 1=a+bi$, then we will get $a=\pm1, b=0$, but why don't we say that $\sqrt 1=\pm1$?
let $\sqrt i=a+bi$ then we will know $\sqrt i=\left(1/\sqrt 2+i/\sqrt2\right)$.
If I apply this method to $\sqrt 1$, then i will get $\sqrt 1=\pm1$, but why don't we say $\sqrt 1=\pm1$?
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Phase Angle of a complex fraction
I'm having a confusion with a problem given, any help will be appreciated.
For example it is given a transfer function,
$G(s)= \frac{(s+20)}{(s+1)(s+100)}$
Substitute $j\omega$ to get the frequency ...
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Why can the $\sqrt{-1}$ not be a real number, but the $\sqrt{i}$ can just be complex? [duplicate]
If I understand correctly, imaginary numbers were invented in order to expand the domain of the square root function into the negative numbers. Curiously though, no such expansion from the complex ...
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Prove if sum of complex numbers = 0 and their magnitudes = 1, then sum of their squares = 0
This problem is from Chapter 6 (Basics of Complex Numbers) of the AoPS Precalculus textbook.
Show that if complex numbers $ w_1+w_2+w_3 = 0$ and $|w_1|=|w_2|=|w_3| = 1$, then $w_1^2 + w_2^2 + w_3^2 = ...
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How to extract variable values from interrelated algebraic equations using programming?
I need help with solving a set of equations that are interrelated.
The equations are:
$$
R(\omega_{s})+j X(\omega_{s})=R_{\mathrm{s}}+\frac{\left(R_{0}+\frac{1}{j \omega_{s} C_{0}}\right)\left(R_{m}+\...
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2
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Computing the limit $a_n = i^{n!}$ as $n \to \infty$
I am trying to compute the limit $a_n = i^{n!}$ as $n \to \infty$, but I am apparently missing some important rule when computing the power of a complex number, as the following doesn't make any sense:...
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Finding the value of an analytic complex function while knowing some of its values
I am trying to deal with this complex analysis question:
Assume that $f$ is analytic within $|z|<1$ and such that $f(\frac{1}{n})=\frac{n^2}{n+1}\sin(\frac{1}{n})$ for $n\in\mathbb{N}$. Find $f(\...
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$\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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0
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18
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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, ...
0
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1
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23
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Confusion about the existence of the antiderivative of this function in this domain.
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
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0
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19
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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 ...
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2
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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 ...
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3
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57
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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?
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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 ...
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4
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99
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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?
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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, ...
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2
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59
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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=...
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(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}^{\...
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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-...
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1
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Polynomial and Complex Roots Problem
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 ...
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