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

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1answer
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Prove that a Möbius transformation $T$ sends the imaginary line to the circle $\{z: |z|=2\}$,

Problem Let $T:\overline{\mathbb C} \to \overline{\mathbb C}$ be a Möbius transformation such that $T(1+2i)=1$, $T(-1+2i)=4$ and $|T(0)|=2$. Show that $|T(bi)|=2$ for all $b \in \mathbb R$. The ...
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2answers
32 views

Cartesian $-10i$ to Polar form

I am trying to convert the following problem to polar form: $$z=-j10.$$ Using this equation, where $|z|=r=\sqrt{x^2+y^2}$ and $\arg z=\theta=\arctan(y/x).$ $$\eqalign{z&=|z|e^{j\arg z}\\ ...
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2answers
53 views

Solve $(2z-1)^5 - i = 0$

Solve $(2z-1)^5 - i = 0$ I started by saying that $(2z-1)^5 = i$ $(2z-1) = \sqrt[5]i$ $z =$ $(\sqrt[5]i +1) \over 2$ $z^5 =$ $(i +1) \over 32$ $z^5 =$ $1 \over32$$ *(i +1)$ From there, ...
3
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4answers
132 views

Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?

Is there a nice geometric, intuitive or picture proof as to why the easily algebraically provable identity $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$ is true? Note I'm not looking for a ...
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2answers
49 views

Product of $n$ complex numbers in rectangular form.

Given a complex number $z_j$ such that $$z_j\in\{a_1+b_1 i,\ a_2+b_2i, \ ...\ ,a_n+b_ni\}$$ is there formula for calculating $$z_1 \cdot z_2 \cdot \dots \cdot z_n =\prod_j z_j?$$ For two complex ...
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3answers
55 views

Complex Numbers and Linear Algebra

Explain why there does not exist a $\lambda $ in the Complex Field such that $$\lambda \left(2-3i, 5+4i, -6+7i \right) = \left(12-5i, 7+22i, -32-9i \right)$$ Can someone help me figure out how to go ...
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4answers
66 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
1
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1answer
44 views

An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
2
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2answers
46 views

Complexes question $(z-1) \over (z+1)$

Given $z \ne -1$. Prove that $(z-1) \over (z+1)$ is an Imaginary number if and only if $|z| = 1$. I tried computing $(z-1) \over (z+1)$ by multiplying like that: $(z-1) \over (z+1)$$(z-1) \over ...
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1answer
24 views

Help on a complex number equality problem

Find the general value of $\theta$ which satisfies the equation $\displaystyle (\cos\theta+i\sin\theta)(\cos2\theta+i\sin2\theta)...(\cos n\theta+i\sin n\theta)=1$ My thoughts: Simplest answer is ...
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2answers
32 views

Proving a Complex number equality

To Prove: If $\displaystyle p=\operatorname{cis}\theta =\cos\theta+i\sin\theta$ and $\displaystyle q=\operatorname{cis}\phi =\cos\phi+i\sin\phi$, then show that $\displaystyle ...
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3answers
128 views

Complex number equality

To Prove: $\displaystyle (\cos\theta +i\sin\theta)^4(\sin\theta-i\cos\theta)=\cos 8\theta+i\sin 8\theta$ My Attempt: $\displaystyle (\cos4\theta ...
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2answers
20 views

Help on Algebraic manipulation of a complex number

If $\displaystyle \frac{1}{x+iy}+\frac{1}{u+iv}=1$; x,y,u,v being real quantities, express v in terms of x and y. My Attempt: $\displaystyle \frac{(u+x)+i(y+v)}{(x+iy)(u+iv)}=1$ $\displaystyle ...
2
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3answers
96 views

Express a complex number in modulus amplitude form

Express a complex number in modulus amplitude form $\displaystyle 1+\sin \alpha +i\cos \alpha $ My Attempt: $\displaystyle r\cos \theta= 1+\sin \alpha $ $\displaystyle r\sin \theta= \cos \alpha $ ...
3
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1answer
77 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
3
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1answer
52 views

Rewriting this complex square root

for some reason I can to figure out how to rewrite this square root. I have: $\sqrt{2+i}$ And I need to rewrite it into: $\frac{\sqrt{2(\sqrt{5} + 2)} + \sqrt{-2(\sqrt{5} - 2)}}{2}$ Can anybody ...
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0answers
32 views

Complex series, radius of convergence

Find the radius of convergence of the series $\sum_{n=1}^{\infty} 4^{n^2}z^n$ and $\sum_{n=1}^{\infty} \dfrac{n!}{n^n}z^n$. For the first one I've applied the root test $\sqrt{4^{n^2}|z|^n}=4^n|z|$. ...
1
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1answer
58 views

Find the 6th root of $-3+4i$ and plot on complex plane

So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is: Find the 6th root of $-3+4i$. I first find the $r$ value which ...
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0answers
22 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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5answers
463 views

What's the importance of a formula for the real and imaginary parts of a complex number?

I've learned that $$\bbox[8px,border:1px solid black]{\operatorname{Re}(z)= \frac{z+\overline{z}}{2} \qquad \qquad \operatorname{Im}(z)=\frac{z-\overline{z}}{2i}} $$ And that in the number $z=a+bi$, ...
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1answer
48 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 ...
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5answers
214 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 $\pi/4$ ...
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1answer
20 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!
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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
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1answer
50 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
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1answer
34 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 ...
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4answers
350 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 ...
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2answers
62 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"?
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2answers
27 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 ...
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2answers
41 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: ...
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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 ...
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0answers
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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 ...
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0answers
29 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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2answers
51 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
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1answer
33 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 ...
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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): ...
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1answer
40 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 ...
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2answers
51 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 ...
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0answers
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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.
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2answers
62 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 ...
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3answers
30 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 ...
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2answers
50 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 ...
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4answers
67 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 ...
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1answer
38 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 $\{Re(z)\}=1$. ...
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2answers
100 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.
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0answers
25 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
19 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
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1answer
180 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,\ ...
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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 ...
1
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1answer
42 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 ...