Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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1answer
19 views

Draw a set of values in complex plane where the complex number $w=1-3i$ is pure imaginary number.

How would you draw a set of values (in complex plane) where the complex number $w=1-3i$ is pure imaginary number? Could this be the solution? If $Rew=0$.
0
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1answer
43 views

How to solve the complex equation? $(x+2yi)^2 = xi.$

How to solve the following complex equation with in less than 60 seconds? $$(x+2yi)^2 = xi.$$ I know how to solve, we have to solve power first then real part equal to real part and imaginary to ...
0
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2answers
83 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
0
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0answers
34 views

Schwarz's Lemma and extensions in complex analysis

I was assigned this problem: (which I here present verbatim) Let $f$ be a holomorphic function of the unit disk unto itself. Prove that $|f'(0)|\le 1$. Isn't it also necessary to assume that ...
1
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1answer
260 views

Form of periodic function involving exponential

I am trying to prove that if the function $f(z)= a_{1}e^{\lambda_{1}z} + ... + a_{n}e^{\lambda_{n}z}$ is periodic of period $T \neq 0$ with $a_{i} \neq 0$ for every $i$, then $\lambda_{i} = 2k_{i}\pi ...
5
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4answers
184 views

Why is it impossible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that $|z_1- z_2|=|z_1-z_3|=|z_2-z_3|=|z_1-z_4|=|z_2-z_4|=|z_3-z_4|$?

A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$. Answer: True B. It is possible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that ...
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2answers
69 views

Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
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2answers
43 views

Prove that $z_1, z_2, z_3, z_4$ are the vertices of a rectangle if and only if…

I have to prove that $z_1, z_2, z_3, z_4$, where $|z_1| = |z_2| = |z_3| = |z_4| = 1$, are the vertices of a rectangle if and only if $z_1z_2z_3+z_1z_2z_4+z_1z_3z_4+z_2z_3z_4=0$ Any help? There is a ...
1
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1answer
50 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
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1answer
36 views

Complex summation simplification

What I'm getting is $$\frac{( \sin (N+1)x - 2^N \sin x)}{(2^N(\sin x - 2))}$$ How do I simplify to the form they have given , please help. I hope it's clear because I don't know Ajax still ...
0
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1answer
44 views

What's the first fundamental form of a regular surface in complex coordinates and how to get it?

Precisely, the first fundamental form of a regular surface is given by $$ds^2=Edx^2+2Fdx\ dy+Gdy^2.$$ What's the form of $ds^2$ in complex coordinates $z=x+iy$.
2
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1answer
30 views

Summation of complex numbers and simplification

By considering $$ \sum_{k=0}^{n-1}(1+i\tanθ)^k\tag{1}$$ Show that $$ \sum_{k=0}^{n-1}\cos(kθ)\sec^kθ=\cotθ\sin(nθ)\sec^nθ\tag{2}$$ Provided $θ$ is not an integer multiple of $\frac{π}{2}$. My take ...
3
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3answers
52 views

System of equations with complex numbers

This might seem quite trivial for people who are knowledgeable in complex analysis, but it is not so much to me. I am trying to find an efficient way to solve the following system of equations: $$ ...
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1answer
48 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
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1answer
51 views

Is this function familiar to anyone?

Consider $$f(z)=\sum_{w\in C}\frac{1}{z-w}$$ Where $C$ is the set of complex integers. What I would like to know is where can I find any information about this function (name perhaps). For instance, ...
2
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1answer
25 views

absolute value of sum of complex numbers squared

is this correct $ \left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-2|a||b|+|b|^2$ Thank you
7
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2answers
633 views

Simplest examples of real world situations that can be elegantly represented with complex numbers

Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
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4answers
1k views

Can one use complex numbers in probability?

I have never thought about using complex numbers in probability. I am examining Bayes Theorem, and attempting to relate it to projective geometry and this question came to mind. I am not talking about ...
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1answer
34 views

Solution to polynomial over complex numbers [closed]

Solve $$z^6 = 8z^3-7z$$ over the complex numbers by considering the roots of complex numbers.
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6answers
295 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
5
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4answers
153 views

$e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?

My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. It seems to me that the formula is a ...
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2answers
76 views

What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
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1answer
37 views

Find the real and imaginary parts of $\sin(\frac{\pi}{2}+i\ln2)$

Find the real and imaginary parts of $$\sin\left(\frac{\pi}{2}+i\ln2\right)$$ Using the double angle formula I have gotten ...
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1answer
88 views

Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$

This is an old qualifying exam problem that I am working on. I would appreciate some help. Thank you. Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$ and ...
21
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8answers
711 views

Is $e^{i\pi}+1=0$ all it's cracked up to be?

While it is beautiful and elegant and all that, isn't it true that Euler's identity is really just an artifact of how we define the radian? I'm speaking of those who say that it's great because it ...
1
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1answer
49 views

Solve $(\frac{z+1}{z})^5 =1$ using fifth roots of unity

$$(\frac{z+1}{z})^5=1$$ Show that its roots are $$-\frac{1}{2}(1+i\cot(\frac{kπ}{5})), k = 1,2,3,4$$ I need to use the five fifth roots of unit, with angles $0,\frac{π}{5}, ...
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2answers
54 views

Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$

I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you. Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for ...
3
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2answers
319 views

The real part treated like an angle in complex vector spaces

In my current lecture I regularly encounter usage of the real part of, say, a scalar product of two vectors similar to angles in classical geometry. For example in Hilbert space theory: Let $H$ be a ...
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1answer
23 views

An inequality on the real part of a square root

I have the following inequality: $\Re(k+z) \geq \Re \sqrt{(k+z)^2-4z}$ where $k$ is real and $z$ complex. Under what conditions on $k$ and $z$ is this inequality true? I suspect that it is true for ...
3
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1answer
83 views

problem about complex integration

The question is to find $\displaystyle\int \frac{z^2-z+1}{z-1}dz$ over $|z|=1$. My solution is : Using cauchy's integral formula we have $\displaystyle f(1) = \frac{1}{2\pi i}\int ...
0
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1answer
40 views

Proof using de Moivre's Theorem

Let $z=\cos\theta + i\sin\theta$ Show that $$1+z = 2\cos\frac{\theta}{2}(\cos\frac{\theta}{2}+i\sin\frac{\theta}{2})$$ I don't even know how to start on this proof.
24
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10answers
2k views

$i^2$ why is it $-1$ when you can show it is $1$? [duplicate]

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with ...
0
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1answer
43 views

De Moivre and trignometry question

I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 ...
0
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2answers
43 views

Finding all the roots from a complex equation

I'm struggling a lot with complex numbers recently. How do I find all the roots for equations like: (1) $\cos z = 3$ (2) $e^{2z} = -e$ (3) $e^z+6e^{-z} = 5$ Thanks
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1answer
800 views

Principal value of complex number

Let $z=-1-i$, find the principal value ? Here $x=-1,y=-1$ therefore $\arg(z)=\tan \alpha=|\frac{y}{x}|=|\frac{-1}{-1}|$ Therefore, $\alpha =\tan^{-1}=\frac{\pi}{4}$ which lies between $0$ and ...
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1answer
44 views

Imaginary unit $i$ is not a limit of a real Cauchy sequence

I saw this in some book once and it has been bugging me. The book had, I think as the first exercise it mentioned, to prove that the imaginary unit $i = \sqrt{-1}$ is not a limit of any real valued ...
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1answer
19 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
0
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1answer
33 views

Let z != -1, which module is 1. Prove that z is presentable z = (1+ti)/(1-ti), where t is real number

Let z != 1, which module is 1. Prove that z is presentable in the following form: $$ z =\begin{align} \frac{1 + ti}{1 - ti} \end{align}$$ where t is a real number So, im guessing i have to ...
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0answers
14 views

What wave does a “complex frequency” correspond to in the Fourier Transform?

The Fourier Transform takes a function $f$, you get another a function $g$. $g$ takes a complex frequency, and returns a sort of relative amplitude of a wave function in $f$. My question is how do you ...
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1answer
71 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
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2answers
36 views

Which is the Hermitian inner product, in terms of conjugate and transpose?

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$ Page 1 of Source 2: $\mathbf{u \cdot v} = ...
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1answer
84 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 ...
0
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2answers
51 views

What is the polar form of -6i?

The module of -6i is 6 (the square root of 36), but $ tan\theta = -6/0$, meaning that the polar form $ 6(cos\theta + isen\theta) $ is also indefinite?
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0answers
32 views

Comparison of two matrix multiplication operations.

I am comparing the below operations: $$ A=\begin{bmatrix} a & 0 & f & g \\ 0 & b & 0 & 0 \\ f & 0 & c & h \\ g & 0 & ...
3
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1answer
63 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
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2answers
220 views

Does it make sense to compare complex numbers in certain circumstances?

I know that $\mathbb{C}$ is an unordered field and that (strictly non-real) complex numbers cannot be 'compared' in the sense that one is less than/greater than another. However, we can compare real ...
5
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1answer
194 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
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2answers
23 views

Re z/z continuous at z=0

How would I show that Re z/z is continuous at z=0? I know that the real value of a complex number equals the sum of the real number and its conjugate divided by two, but I'm not sure where to go when ...
0
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2answers
107 views

How to show that…, $A(C_1)=\{z:|z-1|\leq 1, \theta \in [\frac{\pi}{2},\frac{3\pi}{2}] \}$?

Let $C_1=\{z:|z|\leq 1, \theta \in [\frac{\pi}{2},\frac{3\pi}{2}] \}$ and $A(z)=z-1$. Define $A(C_1)$. How to show that $A(C_1)\neq\{z:|z-1|\leq 1, \arg(z)=\theta \in [\frac{3\pi}{4},\frac{5\pi}{4}] ...
0
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1answer
51 views

How to deal with $\bar{x}$ when solving complex-variable linear equation(s) of x?

The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form \begin{equation} \sum_i a_i x_i=c\ldots\ldots(1) \end{equation} with $a_i,c\in ...