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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120 views

Quadratic Formula for complex variable with real coefficients

I have been upto proving the following $$(\forall x\in \mathbb C, ax^2 + b x + c = 0) \land(a\neq0)\Leftrightarrow (x = {\frac {-b \pm \sqrt{b^{2}-4ac}} {2a}})$$ Due to equality we need to proof bi ...
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54 views

Complex roots of equation $z^\mu=r$

Find the complex roots of equation concerning unknown complex number $z$ \begin{equation} z^\mu=r,\quad \mu>0,r\in\pmb{R} \end{equation} A solution given by a book is to only consider the ...
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2answers
105 views

Solve cos(z) + sin(z) = i, where z is a complex number and i the imaginary unit

So yeah everything is in the title, I tried the trigonometric identity with sin(a+bi) and cos(a+bi) and I tried changing sin(z) and cos(z) for their complex expression, but all to no avail EDIT: ...
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1answer
30 views

Absolute values and inequalities

So I've been trying to solve this one for a few hours and am now out of ideas on how to approach this problem. Here are the inequalities: $$\text{show that if}$$ $$z,w \in \Bbb C$$ $$|z| < ...
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2answers
38 views

Sum of two trig function's identity

We all know that $\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$ But is there an identity for $\sin(x) + z\sin(y) = ?$ Or do I need to figure it out using Euler's formula $\sin(x) = (e^{ix} - ...
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1answer
46 views

Can you graph equations with a negative discriminant? And how do you plot complex numbers both on a 2D complex plane and a 4D complex plane?

I don't understand the relationship between complex numbers and that way they are graphed. The equation I am working with is $2x^{2} - 6x + 5 = 0$ where my two roots are complex solutions: $x = ...
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1answer
50 views

Geometric interpretations of an equality.

I need to prove that for complex numbers $w_1, w_2$ and $w_3$ if: $$\frac{w_2-w_1}{w_3-w_1}=\frac{w_1-w_3}{w_2-w_3}$$ then: $$|w_2-w_1|=|w_3-w_1|=|w_2-w_3|$$ by geometric interpretation of the given ...
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2answers
77 views

For a complex number $c$, how does a plot of $c^{-4},c^{-3}, c^{-2}, c^{-1},c^0, c^1,\dots, c^4$ look like?

I am on the road so can't test it for myself: what would happen if I took a complex number $C = a + bi$ and plotted the following in the complex plane; $$C^{-4}, C^{-3}, C^{-2}, C^{-1}, C^0, ...
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54 views

Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
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1answer
76 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
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36 views

Cauchy-Riemann Equations - why $f'(z_o) = \frac{\partial f}{\partial x}(z_o)$ implies that f is differentiable at $z_o$

I'm trying to understand part b of this proof. The only line I don't understand is the sentence starting with "To prove the statement in (b)..." If someone could clarify why that line is true I ...
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36 views

Computing the tangential and cross components of one quantity using gnomonic projection

I have a spin-2 field given called shape distortion of galaxies as $$\gamma=\gamma_1+i\gamma_2=|\gamma|e^{-2i\phi}$$ where $\phi$ is the orientation angle. If this quantity has been measured on ...
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1answer
35 views

complex functions inequalities plane

Given $w(z)=\frac{i-z}{i+z}$. Find the map w=f(z) of the part of the plane defined by inequalities: $|z|>1$ and $Im(z)>Re(z)$ so far: $|z|>1$ is this area from $Im(z)>Re(z)$ => ...
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34 views

Integrate complex function

In a physics textbook I am working with, the following integral is caluclated: Define the complex elastic modulus as $\overline{M} = M_1 + i M_2$ where $\overline{M}$ is a function of the angular ...
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168 views

Use polar complex numbers to find multiplicative inverse

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(cos(\alpha))+i(sin(\alpha))$. I can do ...
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1answer
68 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
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4answers
33 views

Complex numbers and their modulus

Can we cancel the modulus on complex numbers? For example: If we have $$|x + iy| = |n + im|$$ can we simply ignore the modulus on both sides? Or is that a false assumption?
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231 views

The sum (or difference) of two irrational numbers

So far I that for any irrational number without a real part (that $-n=\overline{n}$) plus/minus any irrational number with the same restrictions equals another irrational number. However, I want to ...
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49 views

Complex numbers product and ratio, prove this relation.

Define a table $T$ as follows: $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = (2+3 i) \sum _{i=1}^{n-1} T(n-i,k-1)+(5+7 i) \sum _{i=1}^{n-1} T(n-i,k) \; \text{else} \; T(n,k) = 0$$ Then take rows ...
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1answer
56 views

Find the set of $z$ which satisfies the given equation

Let $w \to w^{a}$ be the principal branch of the power function defined for $|\mathrm{Arg}(w)| <\pi$. Find the set of all values of $z\in \mathbb{C}$ such that the following identity holds for ALL ...
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73 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
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65 views

Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
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69 views

Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
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1answer
41 views

To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
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2answers
97 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
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29 views

Curves composition with holomorphic function

Statement $(i)$ Let $\gamma:\mathbb R \to \mathbb C$ a $C^1$ curve. Let $v={\gamma}'(t_0)$ the complex number that one obtains from translating to the origin the tangent vector to $\gamma$ at ...
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33 views

Controlling the Sum of a Set of Complex Numbers

Consider a set of N previously fixed angles $\phi_i$. Let $p$ be a positive integer. If $\sum^N_{i=1} e^{ip\phi_i} = 0$, what if any restriction does this place on the value of $p$? If $\phi_i = 2\pi ...
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36 views

Other complex systems

My question would be very short. As we all know, there are complex, quaternion number systems, which are based on multiplication and roots. So, my question is... Is there any other complex number ...
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1answer
83 views

Solving simultaneous equations with complex coefficients using real methods

My circuits analysis textbook teases that there's a way to convert a set of n complex equations into a set of 2n real equations, which can then be solved using any calculator that can solve real ...
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1answer
97 views

Real part of Complex Function

I've this function $$f(k,\theta) = \frac{1}{k}\frac{1}{\cot\delta_0(k) -i }$$ and i know that $k\cot\delta_0(k) = -\frac{1}{a} + \frac{1}{2}r_ek^2 + \cdots$ it is an expansion. How can i get that ...
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37 views

Maximum of $P$ in the disk $|z|=1$ depending on co-efficients

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that ...
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54 views

Complex roots of a complex number [duplicate]

I know how to find the roots to the equation $z^n=w$, for $n \in \mathbb{R\setminus\{ 0\}}$ (by writing $w$ as $re^{i(\theta+2k\pi)}$), and taking the nth root of both sides, which I'm perfectly happy ...
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1answer
25 views

Cauchy-Riemman $w = |z^2|$

So for these types of questions, I can compute the partial differentials for Cauchy-Riemann but then I have trouble seeing/explaining where the function is differentiable? For example with this ...
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1answer
100 views

Using Newton's method to solve a non-linear system of equations over complex numbers

I have a function $f(\bar{z},z)$ mapping from $\mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}^n$, which I would like to find the roots of numerically. Since it is nicely formulated in terms ...
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1answer
103 views

This integral is strange

$$ \int_{C_1}\frac{dz}{z}=\int_0^{2\pi}\frac{-R\sin{t}+iR\cos{t}}{R\cos{t}+iR\sin{t}}dt=\int_0^{2\pi}i\text{ }dt=2\pi i\tag{24.36} $$ Shouldn't it simply be $$\left[\ln(R \cos t + iR \sin ...
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1answer
59 views

Matrices and Complex Numbers [duplicate]

Given this set: $$ S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\} $$ Part I: Why is this set equivalent to the set of all complex numbers a+bi (when both are ...
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46 views

The principle argument of the product of two complex numbers in the second quadrant

I would like some help to prove the following: Show that, if Re $z_1<0$, Im $z_1>0$, Re $z_2<0$ and Im $z_2 >0$, then Arg$(z_1z_2)=$Arg$(z_1)+$Arg$(z_2)-2\pi$. Thanks for any help in ...
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2answers
30 views

locus of complex $z=(λ+3) + i\sqrt{3-λ^2}$

if $z=(λ+3) + i\sqrt{3-λ^2}$, for all real $λ$, then the locus of $z$ is ? Please help. Options are (A) circle (B) parabola (C) line (D) none of these
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When is $\sum_{n,m=-\infty}^\infty \frac{1}{(n\omega_1+m \omega_2)^\alpha}\in \mathbb{R}$?

This came up when reading about elliptic functions, where $\frac{\omega_1}{\omega_2}\notin\mathbb{R}$, and $\alpha>2$ for $$S(\omega_1,\omega_2,\alpha)=\sum_{\begin{matrix} n,m=-\infty\\ (n,m)\ne ...
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73 views

Eigenvalues of complex matrix

I'm taking linear algebra II this semester and the course assumes that students have already covered complex numbers. Unfortunately I take my first analysis course, in which complex numbers are ...
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2answers
70 views

prove the following equation about inverse of tan in logarithmic for

$$\arctan(z)=\frac1{2i}\log\left(\frac{1+iz}{1-iz}\right)$$ i have tried but my answer doesn't matches to the equation .the componendo dividendo property might have been used. where ...
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1answer
36 views

Complex argument and nyquist plot

I'm trying to sketch the nyquist plot of $$\frac{j\omega-1}{j\omega+1}$$ but can't seem to calculate the argument correctly. I think it should be $$\arctan(-\omega) - \arctan(\omega) = ...
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38 views

Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$?

Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$ ? I always thought, that $\Re$ is linear, but if I compute LHS; $\Re(e^{(\lambda ...
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106 views

Evaluation of complex real numbers

The much anticipated math.SE community blog will $\tiny\mathrm{hopefully}$ contain a contribution from Alex Becker with the topic The Complex Real Roots of $x^3-3x+1$, which I'm really looking forward ...
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35 views

To find the value of complex number

If $z$ and $w$ are two non zero complex numbers such that $|zw| =1$ and $\arg z - \arg w = \pi/2$ then conjugate of $(zw)$ =?
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28 views

What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
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25 views

Divisibility by $z-z_0$ if $z_0\in \mathbb{C}$ [duplicate]

I have a problem I'm working on, and I'm just not getting it. Suppose that $z_0\in\mathbb{C}$ is fixed. Show that if $P(z)=c(z^k-z_0^k)$, then there exists a polynomial $Q(z)$ such that ...
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55 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
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63 views

Is this correct: $\ln({(-1)}^{2x-1})=(2x-1)\ln(-1)$?

I would expect the answer to be positive, but it appears otherwise for some values of $x \geq 1$. Here is a simple C++ code that I have used in order to test this: ...
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1answer
48 views

How to solve complex number

how to solve below complex number problem . The points $A,B,C$ represent the complex numbers $z_1,z_2,z_3$ respectively, and $G$ is the centroid of the triangle $ABC$ . If $4z_1+z_2+z_3=0$, ...