Questions tagged [complex-numbers]

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

Filter by
Sorted by
Tagged with
1 vote
2 answers
37 views

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 ...
user avatar
0 votes
6 answers
82 views

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 ...
user avatar
  • 1
0 votes
0 answers
21 views

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 ...
user avatar
0 votes
1 answer
29 views

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=-...
user avatar
  • 2,020
-2 votes
1 answer
58 views

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.
user avatar
0 votes
0 answers
23 views

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\...
user avatar
  • 153
1 vote
1 answer
47 views

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}}\...
user avatar
  • 165
2 votes
3 answers
55 views

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 ...
user avatar
  • 21
0 votes
0 answers
19 views

$\cosh z = 1$, $\sinh z = 1$ [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 ...
user avatar
0 votes
3 answers
52 views

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 ...
user avatar
  • 93
4 votes
2 answers
119 views

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 ...
user avatar
  • 4,139
2 votes
2 answers
123 views

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 ...
user avatar
1 vote
0 answers
18 views

(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 ...
user avatar
  • 153
-3 votes
2 answers
42 views

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?
user avatar
0 votes
2 answers
51 views

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 ...
user avatar
8 votes
1 answer
428 views

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=...
user avatar
0 votes
2 answers
26 views

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|=\...
user avatar
11 votes
1 answer
282 views

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)|...
user avatar
  • 1,328
2 votes
2 answers
49 views

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 ...
user avatar
  • 9,794
5 votes
1 answer
111 views

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 ...
user avatar
  • 1,257
-8 votes
0 answers
33 views

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 $f$ ...
user avatar
3 votes
2 answers
96 views

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 \...
user avatar
2 votes
1 answer
34 views

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\,...
user avatar
0 votes
0 answers
34 views

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 $$ \...
user avatar
3 votes
1 answer
76 views

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 ...
user avatar
0 votes
2 answers
57 views

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 ...
user avatar
  • 153
1 vote
1 answer
64 views

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: $$\lim_{y\rightarrow0^+}\sqrt{-x+iy}=-i\sqrt{x}$$...
user avatar
  • 323
-4 votes
0 answers
49 views

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 $\sqrt 1=\pm1$?
user avatar
1 vote
1 answer
47 views

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 ...
user avatar
1 vote
0 answers
67 views

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 ...
user avatar
1 vote
1 answer
74 views

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 = ...
user avatar
1 vote
1 answer
84 views

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}+\...
user avatar
  • 11
0 votes
2 answers
37 views

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:...
user avatar
1 vote
2 answers
48 views

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(\...
user avatar
0 votes
0 answers
47 views

$\theta$ 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 ...
user avatar
  • 1,591
6 votes
2 answers
128 views

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?
user avatar
-5 votes
1 answer
51 views

Find area of the region in complex plane $\{z+z^2/2: |z| < 1\}$ [closed]

What is the area of the following region in $\mathbb{C}$? $$ \{ z + \tfrac{1}{2}z^2: |z| \leq 1 \} $$
user avatar
  • 1
0 votes
0 answers
18 views

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 $f(3+3i)=g(3+3i)$ ...
user avatar
  • 417
0 votes
0 answers
28 views

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, ...
user avatar
0 votes
1 answer
23 views

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{...
user avatar
  • 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 ...
user avatar
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 ...
user avatar
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?
user avatar
-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 ...
user avatar
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?
user avatar
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, ...
user avatar
-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=...
user avatar
  • 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}^{\...
user avatar
  • 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-...
user avatar
  • 1,234
0 votes
1 answer
63 views

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 ...
user avatar

1
2 3 4 5
354