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

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2answers
115 views

An inequality problem with complex numbers

Knowing that $p$ and $q$ are complex numbers, $|p| < 1$ and $|q|<1$ show that $|\frac{p - q}{1 - q\bar{p}}| < 1$. I've tried to write: $p=x + yi$ and $q=a+bi$ which led me to $x^2 + y^2 + ...
12
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4answers
348 views

Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
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1answer
55 views

What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point Here $A,B$ and $C$ are the point ...
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4answers
1k views

real and imaginary part in $\sin z$ where z is complex

I wanted to know, how can I determine the real and imaginary part in $\sin z$ where $z \in \Bbb{C}$? Well, this is a part of a series of questions comprising the same in $\log z$ and ...
2
votes
4answers
683 views

nth roots of negative numbers

Disclaimer: I know what complex numbers are. Let $x,\space n\in\Bbb R$ What is the complex algebraic solution to $\sqrt[n]{-x}$? Could I have a 'general' formula and a walk through on how to ...
1
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1answer
51 views

How to solve $ \begin{cases} \cos (z_1 +iz_2) = i\\ |z_1|=|z_2| \end{cases} $?

How to solve $ \begin{cases} \cos (z_1 +iz_2) = i\\ |z_1|=|z_2| \end{cases} $? where $z_1, z_2$ are complex variables Rectangular form is convenient for the first equation, and polar form is ...
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1answer
52 views

Boas 17.28 - simplify a complex expression

This is a question from "Mathematical Methods in the Physical Sciences" (Boas, 3rd Ed), Question 17.28: Express the following expression in terms of a hyperbolic function: $$ ...
2
votes
2answers
306 views

Type of solutions for the linear equation AX=B

I have a problem and a proposed solution. Please tell me if I'm correct. Problem: For A,B real matrices, prove that if there is a solution in the complex numbers then there is also a real ...
0
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2answers
94 views

Complex exponents and matrices

If the matrix $A$ is defined as: $$A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & ...
5
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1answer
118 views

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ? How to find if it is convergent or not? Thanks!
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0answers
63 views

Finding $i^{i^i}$. [duplicate]

Express $i^{i^i}$ in the form $a+bi$ where $a,b$ are real. From euler's formula, I get $\ln i=i\pi/2$, which leads to $i^i=e^{-\pi/2}$. Therefore, $\ln i^{i^i}=i^i\ln i=i\pi ...
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0answers
85 views

Solving partial fraction involving complex numbers

This is the image of a part of the problem that I am doing. I understand partial fractions fairly well. To solve for A, you will zero out the other term(s). So, For A, if I let B = 0, I am left with ...
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1answer
123 views

Analog to bisection: Converging on complex roots of a polynomial

I am working on a Perl module that, among other features, will solve all the zeroes of a polynomial. Thus far, I am doing OK for $2$, $3$, $4$th degree using quadratic, Cardano's and Ferarri's ...
5
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1answer
212 views

Good texts in Complex numbers?

I have asked some members on chat about good text to study complex numbers , they recommended for example , "Visual Complex Analysis" by Needham and "complex analysis" by Steins. But, I look for a ...
1
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2answers
995 views

Differentiate a function containing a variable and its complex conjugate

If I have a function of x: $$f(x) = x + \frac K{x^*}$$ Where $x$ is a complex number and $x^*$ is its conjugate. How can I find $f'(x)$ ? My first thoughts are to rearrange: $$f(x) = x + \frac ...
35
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10answers
2k views

What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and ...
2
votes
0answers
176 views

Swap real and imaginary part of a complex number [duplicate]

Say I have some complex number $z=a+ib$ with $a,b\in \mathbb{R}$. Now I want to "swap" its real and imaginary parts, i.e. I want to get $\tilde{z}=b+ia$. While I figure that the appropriate mapping ...
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1answer
1k views

Finding diagonal and unitary matrices

Let $A=\begin{pmatrix} 1 & 1+i\\ 1-i & 2 \end{pmatrix}$ I'm trying to find a diagonal matrix $D$ and a unitary matrix $U$ so that $U^\star AU=D$. (We define $U^*=\overline{U}^t$ ). I ...
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1answer
37 views

form of groups of motions of tessellations

I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says: The triangle and hexagon tessellations have ...
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1answer
59 views

Contour integration of complex number confuses me, still.

Given $f(z) = (x^2+y)+i(xy)$ and we integrate it using the Parabola Contour. For a parabola, $\gamma(t) = t + it^2$. So, $f(\gamma(t)) = 2t^2 + it^3$. What was ...
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1answer
56 views

Complex number inequality

For any $z_1, z_2 \in \mathbb{C}$, is there exist $C>0$ such that $$ 4|z_1|^2 |z_2|^2 + |z_1^2 - z_2^2|^2 \ge C (|z_1|^2 + |z_2|^2)^2 \;\;?$$
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2answers
56 views

Confused between multiple representations of Fourier Series' formula

I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$. Now, suppose ...
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1answer
79 views

Definition of complex number

In many situations (problems as well as solutions) I encounter the complex number $i$ which many times is defined as $i^2=-1$ instead of $i=\sqrt{-1}$, since it is "more preferred". Does anyone know ...
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2answers
325 views

Does $z^i=i^z$ have any solutions, beside $z=i$?

Does this equation have any solutions: $$\large{z^i=i^z}$$ Putting polar form of $z$ is better for LHS, But rectangular form is suitable for RHS ! What to do? Thanks!
3
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1answer
213 views

How to find the bound for this function?

According to journal entitled Certain subclass of starlike function by Gao and Zhou (2007), it was proven that $-\frac{r}{1+tr} \leq Re \{\frac{z}{1-tz} \}\leq \frac{r}{1-tr}$ where $|z|\leq r<1$ ...
11
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9answers
843 views

Solve the equation $z^3=z+\overline{z}$

I have been trying to solve an equation $z^3=z+\overline{z}$, where $\overline{z}=a-bi$ if $z=a+bi$. But I cant find any clues on how to move forward on that one. Please help.
26
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5answers
1k views
0
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3answers
96 views

Can $(-1)^{a+i b}$ be expressed without negative based exponentiation, complex exponentiation, complex logarithms or trigonometric functions?

Can this expression, where $a$ and $b$ are both real, be expressed without negative based exponentiation (i.e. $a^b$ where $a$ is negative), complex exponentiation, complex logarithms or trigonometric ...
5
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1answer
160 views

Do there exist complex algebraic $α,β$ such that $α^β=π$ or $α^β=e$?

Given the algebraic operations and complex exponentiation $(a+bi)^{c+di}$ and logarithm, is it possible to derive $\pi$ and $e$? If one is derivable then so should be the other, as $e^\pi = ...
7
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3answers
203 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
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2answers
637 views

Find the Cube roots of the Complex numbers i+1 and express it in the Argand diagram. [closed]

Find the Cube roots of the Complex numbers i+1 and express it in the Argand diagram.
4
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3answers
170 views

Why don't we define division by zero as an arbritrary constant such as $j$? [duplicate]

Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is ...
4
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2answers
139 views

Question on roots of unity

This may seem absurd but what is wrong with the next reasoning about $n$th roots of unit?. For $k,l\in\mathbb Z$ such that $0 \leq k < l \leq n-1$: $$ e^{2\pi i k/n} = (e^{\pi i})^{2 k /n} ...
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1answer
108 views

Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
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1answer
79 views
5
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0answers
96 views

Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I'm programming a maths environment. I'm computer science, so please excuse any probable ignorance and lack of precision in my question. It seems $i$ and complex numbers were ...
4
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1answer
158 views

A step in the proof of the Riemann Mapping Theorem

Let $\Omega \subsetneq \Bbb C$ be open and simply connected. Let $\overline{\Bbb C}$ denote the Riemann sphere and assume without loss of generality that $0 \in \overline{\Bbb C} \backslash \Omega$. ...
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3answers
335 views

Expressing complex numbers in form $a+bi$

I know that we should express complex numbers generally in the standard form $$a+bi:a,b\in\mathbb{R}$$ Like $4+5i-2=2+5i$. But how do I express complex numbers like $e^{-i\pi/2}$ or $i+e^{2\pi ...
4
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1answer
167 views

Given the norm of a Gaussian integer, how to find the original Gaussian integer?

For $p= a + bi\in\mathbb{Z}[i]$, its norm is $$N(p) = (a + bi)(a - bi) = a^2 + b^2.$$ For example, $N(2+7i) = 2^2+7^2 = 4+49 = 53$. How to find $2+7i$ from $53$? Is there any method?
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4answers
142 views

Is the multiplication of two complex numbers with $|z|=1$ a complex number with modulus 1?

If we have two complex numbers $a, b \in \mathbb{C}$ such that $|a|=1$ and $|b|=1$ is $|a\cdot b|=1$ as well? I am trying to determine if the set $\left(\{z\in\mathbb{C}:|z|=1\},\cdot\right)$ is a ...
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0answers
64 views

How do I express a complex number in a complex base?

I came across an old mathematical paper on the web, published in the 1980s (I can't seem to find it again). The paper was about complex number arithmetic, and it talked about expressing complex ...
26
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7answers
3k views

What does it mean to divide a complex number by another complex number?

Suppose I have: $w=2+3i$ and $x=1+2i$. What does it really mean to divide $w$ by $x$? EDIT: I am sorry that I did not tell my question precisely. (What you all told me turned out to be already known ...
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1answer
69 views

Show that the equation of a line can be given as ℑm(αz+β)=0

I've just started a non-Euclidean Geometry course and the book we are using has a very brief (and not-so-helpful) section on complex numbers that we sort of went over in class. One of the questions ...
2
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3answers
141 views

Show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$

The problem is : if $z$ lies on a circle with diameter having endpoints $z_1$ and $z_2$ then show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$ where $z, z_1, z_2 \in \mathbb{C}$. The angle subtended ...
6
votes
4answers
211 views

Solve $\operatorname{Arg} (z-2) - \operatorname{Arg} (z+2) = \frac{\pi}{6}$

I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$. I know that $$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} ...
0
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1answer
159 views

Fixed Point of $x_{n+1}=i^{x_n}$

For $x \in \Bbb C$, let $f(x)=i^x = \exp(i\pi x)$, where $i^2=-1$. Then find the fixed points for $f$. EDIT: Let for all $n\geq 1$ $$\large a_n=\underbrace{i^{i^{\cdots i}}}_{\text{$n$ times}}$$ My ...
1
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1answer
116 views

How to show that $f(z) = f(i)$

Exercise problem. I do not need a full solution because I am trying to solve myself. Just a hint would be great. Let $f$ be a polynomial with real coefficients. How to show that $f(z) = f(i)$ for ...
2
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1answer
142 views

Uniquely defined function

I am struggling with the following exercise: Let $f:B(0,1) \rightarrow \mathbb{C}$ be a holomorphic function and we have $\forall n \in \mathbb{N}_{\ge 2}: f'(\frac{1}{n})=f(\frac{1}{n})$ then f can ...
5
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2answers
102 views

$w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$

I am stuck on the following problem: Let $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$.Then the triangle in the complex plane with $w_1,w_2,-1$ as vertices ...
1
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1answer
135 views

Limits of complex error and gamma functions in the complex plane?

What are the following one-sided limits in the complex plane (in the form $x+iy$): For the complex error function: $\lim_{x \to 0^+, y \to 0^+}\text{erf}\left(x+iy\right) = $ $\lim_{x \to +\infty, ...