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

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2
votes
1answer
46 views

Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an ...
6
votes
5answers
199 views

How do i find $(1+i)^{100}?$

How do I find $(1+i)^{100}$ without expanding $(1+i)$ 100 times? Is there a quicker way to do this? The hint was to find the modulus and argument of $1+i$ which I've got as $\sqrt{2}$ and $\pi/4$ ...
0
votes
1answer
19 views

Quadratic factor to complex numbers

How to convert this quadratic factor to complex number form? (With steps please) Reference: $Z = a + bi$, $i = \sqrt{-1}$ $$-3 + \frac{\sqrt{-12}}{2}$$ Thanks!
0
votes
0answers
41 views

Why Does $e^{ix}=\cos(x)+i\sin(x)$? [duplicate]

Something I've always wondered, but never had a good answer too (I accept there may not be one). I fully understand how to derive this, so I'm not looking for an analytic proof. But rather I cannot ...
3
votes
1answer
49 views

Finding Möbius transformations that satistfy certain conditions

Problem Find Möbius transformations that send $(i)$ the circle $|z|=2$ to $|z+1|=1$, and $-2$ to $0$, $0$ to $i$. $(ii)$ the upper half-plane $Im(z)>0$ to $|z|<1$ and $\lambda$ to $0$ (where ...
0
votes
1answer
33 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
1
vote
4answers
344 views

Is it true that “there is no such thing as the square root of minus one”?

Is the statement "there is no such thing as the square root of minus one" a true statement? It seems to me that we need to be careful about the word "the" as it appears in the statement. If we see it ...
0
votes
2answers
61 views

Complex numbers: How to solve the “contradiction”? [duplicate]

$$-1 = i\cdot i = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$$ $$-1 = 1$$ Obviously, something is wrong here, but I can't put my finger on it. How to solve this "contradiction"?
1
vote
2answers
24 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
1
vote
2answers
38 views

Sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1 \}$

My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. I substituted $z$ for $a+bi$, but did not get anywhere: ...
0
votes
0answers
21 views

Is alpha and endomorphism of C considered as avector space over R? Is it an endomorphism of C considered as avector space over itself?

Let alpha:C$\to$C be the function defined by alpha:a+bi$\to$ -b+ai.(1) Is alpha and endomorphism of C considered as a vector space over R?(2) Is it an endomorphism of C considered as a vector space ...
1
vote
0answers
26 views

Usefulness of alternative constructions of the complex numbers

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One ...
1
vote
0answers
28 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 ...
0
votes
2answers
47 views

Finding the modulus of a complex number that satisfies a polynomial relation

Consider $z\in\mathbb{C}$ such that $z^2-2z+3=0.$ Find the modulus of $$f(z)=z^{17}-z^{15}+6z^{14}+3z^2-5z+9$$ My attempt: $z^2-2z+3=0\Leftrightarrow\left[ ...
2
votes
1answer
29 views

Cross ratio and symmetric points exercise

Problem Let $C$ be a circle or a line belonging to $\overline{\mathbb C}$ and let $z_2,z_3,z_4$. Two points $z$ and $z^*$ are said to be symmetric with respecto to $C$ if ...
4
votes
2answers
38 views

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$ I went wrong somewhere, this is what I have so far (this is in polar): ...
0
votes
1answer
36 views

Möbius transformations on $\space \overline{\mathbb R}$

Prove that a Möbius transformation $T(z)=\dfrac{az+b}{cz+d}$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ if and only if it can be written with real coefficients. If it can be written with ...
0
votes
2answers
50 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
0
votes
0answers
12 views

what is deformation of a contour and Fourier inversion in general?

I would like to to know how one can do contour deformation and Fourier inversion in general? I shall be very thankful to you if some one explain it with the help of example.
1
vote
2answers
59 views

Solve for x if $z$ is a complex number such that $z^2+z+1=0$

I was given a task to solve this equation for $x$: $$\frac{x-1}{x+1}=z\frac{1+i}{1-i}$$ for a complex number $z$ such that $z^2+z+1=0$. Solving this for $x$ is trivial but simplifying solution ...
1
vote
3answers
29 views

Related to the construction of $\Bbb C$ (generalisation)

To construct $\Bbb C$, we consider $\Bbb R^2$ endowed with the operations: $$\begin{align} (a,b) + (c,d) &:= (a+c, b+d) \\ (a,b) \cdot (c,d) &:= (ac - bd, ad+bc)\end{align} $$ then write ...
1
vote
2answers
46 views

Trigonometric equation with complex numbers

Let $x$, $y$, and $z$ be real numbers such that $\cos x+\cos y+\cos z=\sin x+\sin y+\sin z=0$. Prove that $\cos 2x+\cos 2y+\cos 2z=\sin 2x+\sin 2y+\sin 2z=0$. Starting with the given equation, I got ...
0
votes
4answers
63 views

complex numbers quadratic equation question

how to solve $z^2 +3|z| = 0 , z$ complex ? treating the complex number as $a+bi $ or anything similar didnt help much...also solving like simple algebric equations also didnt prove effective and ...
3
votes
1answer
37 views

Möbius transformation: proving the image of the unit circle is a line

Problem 1) Find the Möbius transformation which maps the points $0,i,-i$ to $0,1,\infty$ respectively. 2) Prove that the image of the circle centered at $0$, of radius $1$ is the line $\{Re(z)\}=1$. ...
0
votes
2answers
88 views

Why is this wrong (complex numbers and proving 1=-1)?

$$(e^{2πi})^{1/2}=1^{1/2}$$$$(e^{πi})=1$$ $$-1=1$$ I think it is due to not taking the principle value but please can someone explain why this is wrong in detial, thanks.
1
vote
0answers
24 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 ...
1
vote
1answer
18 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 ...
5
votes
1answer
177 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
1
vote
1answer
12 views

Images of the stereographic projection's inverse

I am trying to solve a problem which states: Let $\phi: \bar{\mathbb C} \to S^2$ be the inverse function of the stereographic projection Calculate $\phi(Re(z))=0$ and $\phi(Im(z))=0$. I can guess ...
-3
votes
0answers
44 views

Complex numbers quesiton

z1 z2 z3 are three points on the gauss plane and create a triangle ; it is known that z1 = 3sqrt3 +3i, z2 = z1/cis240 deg , z3 = z2cis60 deg . given this data show that the triange is a staright ...
1
vote
1answer
40 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 ...
1
vote
0answers
34 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 ...
1
vote
3answers
83 views

What is the real and imaginary parts for the complex function $f(z)=z^z$

I know: $f(z)=z^z =|z|^ze^{iz\theta} $ and $=|z|^z(\cos(zθ) + i\sin(zθ))$ But how do I continue to get the results for $\Re(z^z)$ and $\Im(z^z)$? $$\text { }$$ Thanks.
1
vote
0answers
23 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 ...
1
vote
2answers
76 views

Operations with complex numbers to give real numbers

If: $|z|=|w|=1$ $1 + zw \neq 0$ Then $\dfrac{z+w}{1+zw}$ is real. How can prove that.
0
votes
3answers
29 views

Product of the n roots of the unity can be $(-1)^{n-1}$

I need to prove that the product of the n roots of the unity can be $(-1)^{n-1}$. If i make $1^n=z$, z can be $ cis(\dfrac{k \cdot 2\Pi}{n})$ with k=0, ... , n-1 Now i need to prove that ...
4
votes
1answer
45 views

relationship between complex numbers

Consider the following: Two equilateral triangles inscribed in a circle. The vertices of the large triangle are the geometric images of the three cubic roots of $z$ (a complex number). The small ...
0
votes
1answer
28 views

Principal values of complex functions

How do I find the principal value of the following: $\log(1-\sqrt{2i})$ And hence of $(1-\sqrt{2i})^i$ Also how do I write $z=1+i$ in polar form and find its roots? I find these very confusing ...
0
votes
1answer
38 views

Proving the Mandelbrot set is bounded

I am trying to prove that the Mandelbrot set defined as the set $\mathcal M$ of complex numbers $c$: the recursive sequence defined as $$z_0=c, \space \space \space z_{n+1}={z_n}^2+c$$ is bounded. ...
1
vote
1answer
23 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 ...
6
votes
1answer
62 views

Prove that $\cos(z)$ and $\sin(z)$ are surjective over the complex numbers. [duplicate]

I have an exercise that says: (a) Prove that $\cos(z)$ and $\sin(z)$ are surjective functions from $\mathbb C \to \mathbb C$. (b) Find the solutions of the equation $\cos(z)=\dfrac{5}{4}$. As far ...
2
votes
4answers
588 views

Is the power of complex number defined yet?

Let $z$, and $b$ be two complex numbers. What is $$f_b(z)=z^b.$$ If I write it like this: $$ \left(re^{i\theta}\right)^{b}=r^{b}e^{ib\theta}. $$ Would this even make sense? Wolframalpha gives me ...
0
votes
1answer
72 views

How to find the roots of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ using trigonometric methods

Can all the roots (real or complex) of $x^6 + x^5 +x^4 + x^3 +x^2 + x = n$ be found using trigonometric methods? Many thanks to all of answers.
1
vote
2answers
53 views

How to prove that :$\prod_{k=0}^{n-1}e^{\frac{2\pi i k}{n}}=(-1)^{n-1}\;\;\; n\in\mathbb{N}^*$

Can someone tell me how to prove the folowing equalty : $$\prod_{k=0}^{n-1}e^{\frac{2\pi i k}{n}}=(-1)^{n-1}\;\;\; n\in\mathbb{N}^*.$$ Thanks in advance.
0
votes
1answer
37 views

Determining Complex number

Let $ z_1,z_2$ $\in$ $\mathbb{C}$ and A,D their respective images in the complex plane. let B be the image of $z_1³$ in the complex plane. A is in the first quadrant and B is in the second quadrant. ...
0
votes
1answer
38 views

Proving $\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$

Prove that $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}=\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$$ I tried to use $$ ...
2
votes
4answers
91 views

Proving by calculation that $\arg(-2) = \pi$

The fact that it is true, seems very obvious, if one draws the complex number $z = (-2 + 0i)$ on the complex plane. The angle is certainly 180 degrees, or pi radians. But how can this be proven by ...
1
vote
2answers
44 views

Writing $e^{i\theta}(e^{in\theta}-1)/(e^{i\theta}-1)$ in $(a+i b)$ form

How to write: $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}$$ in $$(a+i b) $$ $$ ?$$ I tried to multiplicate by $$e^{i}$$ (the numerator and ...
4
votes
4answers
54 views

Proof of trigonometric identity $\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$

I was given a task of proving the following identity: $$\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$$ I am not looking for a solution, just some kind of a hint to start ...
0
votes
1answer
16 views

Complex number Q1

For $z\in \mathbb{C}\backslash \{i\}$. How can i go from this line : $|z|^2-Re[(1-i)z]=0$ To that one : $$\left|z-\dfrac{1+i}{2}\right|^2=\left|\dfrac{1+i}{2}\right|^2=\dfrac{1}{2}$$ indeed, ...