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

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Complex Numbers - Sketching on Argand Diagram

Sketch the subsets of the Argand diagram - Draw near labelled sketched to indicate each of the subsets of the Argand diagram described below. $\{z: |z|\ge 1\text{ and ...
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1answer
71 views

Solve the following complex number: $\frac{1 + i\tan \theta}{1 - i\tan\theta}$

How do I solve the following complex number? $$\frac{1 + i\tan \theta}{1 - i\tan\theta}$$ I know how to solve arithmetic problems with complex numbers, but this is the first time I have a function ...
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2answers
154 views

Rewriting $x^3-3xy^2+2xy+i(-y^3+3x^2y-x^2+y^2 )$ in terms of $z$, with $z=x+yi$

How do I write $f=u+iv$ with: $u=x^3-3xy^2+2xy$ and $v=-y^3+3x^2y-x^2+y^2 $ in terms of $z$ with $z=x+yi$?
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797 views

Circles in Complex Planes

Points on the circle centre C and radius r are given by the equation $|Z-C|=r$ or $(Z-C)(\overline{Z}-\overline{C})=r^2$. Where $Z = x + iy$. When multiplied out, I understand that we have ...
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Proving $|\sum _{ k=1 }^{ n }{ z_{ k } } |=\sum _{ k=1 }^{ n }{ |z_{ k }| } \Longleftrightarrow \arg(z_1)=\arg(z_2)=\cdots=\arg(z_n) $

Suppose $z_1,\dots, z_n$ are $n$ elements from $\mathbb{C}^*$. How can I prove that $|\sum _{ k=1 }^{ n }{ z_{ k } } |=\sum _{ k=1 }^{ n }{ |z_{ k }| } \Longleftrightarrow ...
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Trigonometry with complex numbers [closed]

Express $(\cos(x))^5$ in terms of cosines of multiples of $x$. I've racked my brains for ages on this one! No notes to help me out, and I've failed to find any help online.
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Complex conjugate to the power proof.

How can I proof that: $$(z^n)^* = (z^*)^n$$ Where: z is a complex number, n is a positive whole number * is the complex conjugate
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Taking derivatives of exponential function

Beware, this question might be silly and may contain mathematical fallacies. $$ d/dt(e^{jwt}) = jwe^{jwt} $$ $$ d/dt(e^{j \pi t}) = j \pi e^{j \pi t} $$ $$ d/dt(e^{j 180 t}) = j 180 e^{j 180 t} $$ ...
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1answer
35 views

Quadrant problem

$(-1+i)^{\frac{1}{3}}$ here, $\tan\theta=-1$ so, $\theta=\tan^{-1}(-1)=\tan^{-1}(\tan(-\frac{\pi}{4}))=\tan^{-1}(\tan(\pi-\frac{\pi}{4}))=\pi-\frac{\pi}{4}$ My question is why can't i write ...
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2answers
61 views

How to get to the first step of solving $z^2=i$?

How do I get from $z^2=i$ to $z=x+iy$? Is it a rule you use to solve the equation in general or specifically for this equation? (I don not understand the step from $z^2=i$ to $z=x+iy$) Thank you!
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4answers
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How to solve a complex polynomial?

Solve: $$ z^3 - 3z^2 + 6z - 4 = 0$$ How do I solve this? Can I do it by basically letting $ z = x + iy$ such that $ i = \sqrt{-1}$ and $ x, y \in \mathbf R $ and then substitute that into the ...
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93 views

Re-write arguments between $(-\pi,\pi]$ - why

I have a complex number answer with an argument of $4\pi/3$ and the example said to make the argument a number between $(-\pi,\pi]$ however I don't understand what that means or why we do it (the ...
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1answer
61 views

Pfister's 16-Square Identity and the norm of sedenions

Consider the sequence of numbers: complex numbers $\Bbb C$, quaternions $\Bbb H$, octonions $\Bbb O$, and sedenions $\Bbb S$. The Brahmagupta-Fibonacci 2-Square identity implies that the norm of the ...
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1answer
162 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 ...
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Reference request for derivatives of complex functions

I have been searching for reference for derivatives of complex numbers. All I found so far were texts that were too convoluted for me to grasp. I was (and still am) searching for a reference that is ...
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173 views

Letters for complex numbers

Suppose that I am writing a proof or some other piece of mathematical writing, and wish to introduce $n$ distinct complex numbers, for some positive integer $n$. What are the complex numbers called? ...
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135 views

What exactly is the complex plane, and how is it useful?

A lot of functions are defined on the complex plane, like the Gamma function: the Lambert W function, etc. But I have no idea about what the complex plane means and how it's useful, or just ...
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84 views

Simplification of product of complex numbers

I look for a closed formula to the expression $$\prod_{k=1}^{n-1}\left(e^{\frac{2ik\pi}{n}}-1\right)$$ Any suggestion is welcome. Thanks.
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Two types of solution to the differential equation

I have read that we can have two solutions to the second order DE below, where $W$ and $W_p$ are constantants and $\psi$ is a function of $x$: $$\frac{d^2\psi}{dx^2} = -(W-W_p) \psi $$ (a) If we ...
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80 views

Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where $x,y$ complex numbers?

Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where x,y are complex numbers? Or is it simply solved by observation, and so the answer is $x=-i$ and $y=1$? Thanks in advance.
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1answer
50 views

constructing infinitely long path

I would like to show that $\displaystyle \left|\int_C{{z^{-4}}} ~dz \right| \le 4\sqrt2$ where $C$ is ANY path between $z = i$ and $z = 1$. Show that a path $C$ where the length of that path is ...
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62 views

How to evaluate $|z-1/2|$ if $|z| \leq 1$?

How to evaluate $|z-1/2|$ if $|z| \leq 1$? I got: $|z| \leq 1 \Leftrightarrow 1/2 \leq |z-1/2| \leq 3/2$.
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Set of points that fulfill a formula

Suppose $u=\frac{z(1+i)-i}{z+1}$ as $z\in \mathbb{C} \setminus\{-1\}$ What is the set of points $M(z)$ for which $u$ is a real number? What is the set of points $M(z)$ for which $u$ is pure ...
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399 views

equality of two complex numbers

For $z\in { C }, \Re (z)\neq 2$ we have $F(z)=\frac { 4-z\overline { z } }{ 4-z-\overline { z } }$. I'm trying to prove the equality between the modulus of these numbers without using the ...
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493 views

Points on complex plane

Describe the set of points $z$ in the complex plane that satisfies the following: a.) $|z-1| + |z+1| = 7$ b.) $|z| = 3|z-1|$ For a, I know that it has the property that their distance ...
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87 views

A question about a complex variable function

My question is about the function $f(z)=e^{-z^2}$. Is it everywhere continous? Holomorphic? Thanks, Dan
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1answer
40 views

what is the difference - sorry for over-simplicity

i am asking too simple question, sorry for that. what is the difference between these two imaginär numbers? $\operatorname{Im}(| \sqrt2+3i|^2)$ vs. $\operatorname{Im}((\sqrt2+3i)^2)$ $| ...
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213 views

Finding the locus

Let “Z” be a complex number then the locus represented by |Z-1| + |Z+2| = 2 is? How could we show effectively that no complex number satisfies this?
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132 views

How can I verify that cos z dz is exact?

The only way I know how to verify something as an exact differential is when it is in the form $P(x,y) dx + Q(x,y) dy$. The book's definition states that such a differential is exact if there exists ...
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730 views

Finding the roots of complex numbers

$(2i)^{1/2}$ $(1-\sqrt{3}i)^{1/2}$ $(-1)^{1/3}$ $(-16)^{1/4}$ How can I find the roots of the complex numbers above?
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1answer
182 views

Complex n-th root question

Let $m$ and $n\neq0$ be any two integers.Show that $z^{m/n}=\left(z^{1/n}\right)^m$ has $n/(n,m)$ distinct values, where $(n,m)$ is the greatest common divisor of $n$ and $m$. Prove that the sets of ...
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3answers
83 views

Circle in a complex plane.

Let $C$ be a circle in the complex plane, and let $x$ be a fixed, non-zero complex number. Prove that $\{xz : z \in C\}$ is also a circle. I would really appreciate any help that would get me ...
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1answer
100 views

Linear independence of $\{\exp(b_{n}z):n\in\mathbb N\}$

I have the following question: Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for some complex point $z\in\mathbb Z$. Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ ...
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515 views

Solving absolute value inequalities.

Prove: If $z,\alpha \in \mathbb{C}$ with $|z|<1, |\alpha|<1$ then $\dfrac{|z|^2+|\alpha|^2}{1+|\alpha z|^2}<1$
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628 views

Rectangular form of a complex number?

Why does rectangular form serve as an accurate description of a complex number? Why not $a * bi$(multiplication) or another operation? Why does addition describe the relationship between the real part ...
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702 views

Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
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Complex conjugate of function

I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants. I am trying to find the probability density, so I need to find the product of $\psi$ ...
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236 views

Complex Numbers task

I can't solve this set of equations, please help me. $$(1+i)z_1 + (1-i)z_2 = 1+i$$ $$(1-i)z_1 + (1+i)z_2 = 1+3i$$
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Finding centre and radius of circle

Let $a,c \in \mathbb R$ with $a \neq 0$, and let $b \in \mathbb C$. Define $$S=\{z\in \mathbb C: az\bar{z}+b\bar{z}+\bar{b}z+c=0\}.$$ a. Show that $S$ is a circle, if $|b|^2 > ac$. ...
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1answer
176 views

How to prove this easy problem?

For $0<\alpha<1$, we define $\Omega_\alpha$ to be the union of the disc $D(0;\alpha)$ and the line segments from $z=1$ to points of $D(0;\alpha)$. Now, let $r=|z|$. Given $0<\alpha<1$, ...
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101 views

How does a $y(t)=$plot of this exponential function looks like?

I've tried to plot the y(t) function, but there are complex numbers and I don't know how to plot it. I've been looking for hyperbolic functions transformation, but I didn't figured how to do this. My ...
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89 views

Correct representation of $\omega^2$ in Euler form?

I have been trying to write $\omega^2$ in Euler form, first $$\omega^2 = \frac{-1}{2}-\frac{\sqrt{3}i}{2} ,$$ hence $|\omega^2| = 1$ and $\arg(\omega^2) = -\pi + \frac{\pi}{3}$ as $\omega ^2$ lies in ...
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1answer
20 views

How do I write this complex number in exponential form?

$$ -4 - i 16\sqrt{5}$$ Example: I know we can write $-8-i8\sqrt{3}$ as $16e^{i(-2\pi/3 + 2k\pi)}$ where $k = 0,\pm1, \pm2,....$
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1answer
53 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
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35 views

Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
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1answer
39 views

Simplifying $\frac{8-i}{2+3i}$ to standard form

I am just learning about "imaginary" numbers and having trouble understanding it all. I'm supposed to write the following complex number in standard form. $$\frac{8-i}{2+3i}$$ How can I start?
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Could someone point me in the right direction for this complex analysis equation?

I'm supposed to show that the maximum value of $|z^2+1|$ on the unit disk $|z|\leq1$ is 2. My teacher's hint was "triangle inequality". I've been racking my brain how to tie the triangle inequality ...
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44 views

How would one prove this flavour of the triangle inequality?

I have to prove $|z_1 - z_2| \leq |z_1|+|z_2|$ where $z_1,z_2$ are in $\mathbb{C}$. What I wrote down is: $$|z_1| = |z_1+z_2-z_2| \geq |z_1-z_2|-|z_2|\implies |z_1|+|z_2|\geq |z_1-z_2|,$$as desired. ...
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3answers
48 views

How to compute $(1 − i \sqrt{3})^3\cdot(1 + i)^2$ using the trigonometric form of complex numbers?

I need to compute it using the trigonometric form of complex numbers: $$(1 − i \sqrt{3})^3\cdot(1 + i)^2$$ I computed it using the standard method: $-16i$
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1answer
22 views

Complex numbers working with real definition

suppose that $z = (x,y)$ and $z^2 = (-1,0)$. Show that $z = i$ or $z = -i$ Now in this form the ordered pairs are defined for the real numbers, Sonmy idea was by definition of multiplication: $Z^2 ...