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

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2
votes
1answer
22 views

If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then either $f\equiv0$ on $\Bbb C$ or $f(z)\not =0$ for all $z\in \mathbb C$.

Let , $f,g:\mathbb C\to \mathbb C$ be analytic such that $g(z)\not =0,\forall z\in \mathbb C$. If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then prove that either $f\equiv0$ on $\Bbb C$ or ...
2
votes
1answer
33 views

Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns?

When computing returns in finance geometric means are used because the return time series of a financial asset is a geometric series: $\mu_r = \sqrt[T]{\prod_{t=1}^T r_t}$ where the return is computed ...
0
votes
2answers
26 views

Where does this equality come from? complex numbers rewritten

http://mathfaculty.fullerton.edu/mathews/c2003/ComplexSequenceSeriesMod.html See example 4.2. in above. They have $z_n = (1+i)^n$ and then they've rewritten that to a familiar $a_n+ib_n$ form ... ...
0
votes
1answer
11 views

Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
0
votes
0answers
33 views

Solve the complex euqtions

I have a question from complex calculus. How to solve this two equations: a) $$ sin(z)=2015 $$ I know that sin(z) equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
-3
votes
1answer
31 views

Prove that $x^3+x^2+x$ is a factor of $(x+1)^n-x^n-1$ using complex numbers [on hold]

This question is given in my book under the complex number chapter but I can't understand how to solve it using complex numbers. It is given that $n$ is an odd integer greater than 3, but $n$ is ...
3
votes
6answers
341 views

Expansion of complex equation.

Find the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than ...
2
votes
5answers
51 views

Solving $\cos z = i$ for $z$

Solve $\cos z = i$ for $z$. What I have tried: $$\cos z = i$$ $$\frac{e^{-zi}+e^{zi}}{2}=i$$ $$e^{-zi}+e^{zi}=2i=2e^{\frac\pi 2 + 2\pi k},\quad k\in \Bbb Z$$ I would take logs, but then I would ...
5
votes
5answers
149 views

Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?

I am an $8^{th}$ grader that is taking Algebra I. But nearly everyday I try to learn things outside of what I am learning in class. Quite a while ago I discovered that $e^{i\pi} = -1$. This ...
0
votes
0answers
21 views

Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
0
votes
0answers
18 views

Harmonic Function Cauchy implication

Let $b$ be harmonic real valued on unit disk. Then I wish to prove that $\int_\alpha b =0$. I know that there exists $f$ holomorphic such that $\Re(f)=b$, and I know from Cauchys result that ...
0
votes
2answers
23 views

solving the limit $\lim_{n\to \infty}\sum_{k=1}^n|e^{(2πik)/n}-e^{(2πi(k-1))/n}$|

$$\lim_{n\to \infty}\sum_{k=1}^n\left|e^{(2πik)/n}-e^{(2πi(k-1))/n}\right|$$ i can solve it geometrically. but is there any way to solve it using Euler's formula ?, the answer will be one of these ...
1
vote
1answer
24 views

Schwarz Lemma, an onto map with $f'(0)>0$ is the identity

Let $f$ be $1-1$ holomorphic on unit disk onto itself. It satisfies (a) $f(0)=0$, (b) $f'(0)>0$. We need to prove that $f(z)$ is equal to $z$. I am stuck here, because I can prove using Shcwarz ...
0
votes
1answer
37 views

Unique properties of pure Imaginary numbers?

Are there any non trivial properties unique of the imaginary numbers? By trivial I mean stuff like $\bar a=-a$.
-1
votes
2answers
91 views

The real part of the sum $(i-1)+(i-1)^2+(i-1)^3…+(i-1)^{2013}$?

I'm not sure how to go around this one. Factorizing doesn't seem to work and there isn't a clear pattern to work by that I see. EDIT: So apparently I need to add context and stuff. I removed the ...
0
votes
1answer
18 views

Prove that if $Re(z)>0$ then $|z+\sqrt{z^2-1}| \ge 1$

This is probably a very basic question in complex numbers. First define $\sqrt{w} := \sqrt{|w|}e^{i(Arg(w)/2)}$ where Arg is the principal argument function. Prove that if $Re(z)>0$ then ...
0
votes
0answers
25 views

Complex Geometry Problem

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find [$PA_1^2 + PA_2^2 + \dots ...
2
votes
3answers
236 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
5
votes
4answers
60 views

Find all solutions to the following equation: $x^3=-8i$

Find all solutions to the following equation: $$x^3=-8i$$ I found the modulus, $$r=8$$ $$\operatorname{arg}(x)=\arctan(-8/0)=-π/2+2πk$$ By De Moivre's Theorem: ...
0
votes
0answers
17 views

Algorithm to find value where complex numbers meet on the unit circle. [on hold]

I'm trying to find the value at which 4 points are meeting on the unit circle. These points are eigenvalues of the translation operator $T$. By varying $\lambda$ the eigenvalues change. Background: ...
0
votes
1answer
34 views

complex numbers- how do I prove the following statement? [duplicate]

given: $$ \left | z_{1}\right |=\left | z_{2}\right |=...=\left | z_{n}\right |=1 $$ How do I prove: $$ ...
1
vote
1answer
19 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
0
votes
0answers
29 views

Does rule of three make sense for other than real numbers?

I'm currently working on a software tool which can make calculations based on the rule of three. I can make it more simple and just support real numbers, or I can make it "universal" to support ...
0
votes
5answers
40 views

Complex Numbers Question, IIT JEE [2006]. Please tell me whether I solved it properly?

$Q.$The value of $\sum\limits_{k=1}^{10}(\sin{\frac{2k\pi}{11}-i\cos\frac{2k\pi}{11}})$ is-? I solved it like this- $\frac{\sum\limits_{k=1}^{10}(\cos{\frac{2k\pi}{11}+i\sin\frac{2k\pi}{11}})}{i}$ If ...
0
votes
0answers
34 views

Equating eigenvalues of Hermitian matrix and correlating symmetric/antisymmetric matricies

I have a matrix $AH$ which is created by adding $AS$ and $i*AA$, which are the symmetric and antisymmetric components of the real matrix $A$ So $AS=(A+A')/2$ $AA=(A-A')/2$ $AH=AS+i*AA$ AH has ...
2
votes
1answer
53 views

Area of triangle formed by angle bisector, altitude and median

Question:- Given a triangle ABC with side length a, b and c. Calculate the area of a triangle in terms of a, b and c formed by angle bisector from vertex A, altitude from vertex B and median from ...
0
votes
2answers
24 views

Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
2
votes
3answers
37 views

How is this step done? $\left|\frac{i\overline{z}}{2} -\frac i2\right|=\frac{|z-1|}{2}$

Absolutely everything makes sense other than what is in red. How is this step completed? Let us show that if $f(z)=\dfrac{i\overline{z}}{2}$ in the open disk $|z|\lt 1$, then$$\lim ...
1
vote
1answer
26 views

Finding eigenvalue and eigenvectors of a matrix containing an imaginary number

How do you solve for the eigenvalues given the matrix? \begin{matrix} i & -2 \\ 1 & 0 \\ \end{matrix} I know how to get the characteristic polynomial Ca(X); X^2 - ...
3
votes
2answers
78 views

how can I simplify this $\sqrt{i}+\sqrt{2i}+\sqrt{3i}$

Is there an easy way to simplify the $$\sqrt{i}+\sqrt{2i}+\sqrt{3i}$$
0
votes
1answer
43 views

Cube root of $\omega$

What is the cube root of $\omega$? where $\omega$ is the non-real cube root of $1$. I got the cube root of $\omega$ in terms of $\omega$! I took cube root of $\omega$ as $x+iy$ then cubed it. I got ...
2
votes
3answers
59 views

Find all values of $\sqrt[4]{-1+i}$

Okay. I know how to solve for all values of $\sqrt{-1} $ but $\sqrt{-1+\iota} $ confuses me a bit. I got the value of r to be $\sqrt 2 $ I ended up with this: $ z_k = ...
1
vote
4answers
73 views

is 1 greater than i?

I'm not sure this question even makes sense because complex numbers are a plane instead of a line. The magnitudes are obviously the same because i is a unit vector, but is there any inequality you can ...
1
vote
2answers
22 views

$Im(\cosh z)=\sinh x\sinh y$ and $|\sinh z|^2=\sinh^2(z)+\cosh^2(z)$

I have problems in two issues of complex variables ... 1) Prove that $Im(\cosh z)=\sinh x\cosh y$, if $z=x+iy$. I tried to expand $\cosh z= ...
2
votes
1answer
64 views

Prove that $f$ has a simple pole at $z=0$

Let, $f:\{z\in \mathbb C:0<|z|<1\}\to \mathbb C$ be analytic such that $n\le |f(1/n)|\le n^{3/2}$ for $n=2,3,...$. Assume that $z^2f(z)$ is bounded in $|z|<1$. Show that $f$ has a pole of ...
0
votes
6answers
66 views

Find the three roots of $z^3 = -i$ in the form $a+ib$. [on hold]

Find the three roots of: $$z^3 = -i $$ in the form $a+ib$.
1
vote
0answers
24 views

Locus of complex numbers.

Question Let $P(x,y)$ be the point on an Argand diagram representing the complex number $u=x+iy$ and satisfying the equation \begin{align*} \vert u \vert=k\vert u+a\vert, \end{align*} where $k$ is ...
0
votes
1answer
35 views

$|z^i|<e^\pi,\;\;\forall z\in\mathbb{C}-\{0\}$

Good morning people ... Do you have any idea to help me prove that $$|z^i|<e^{\pi}$$ for $z\in\mathbb{C}-\{0\}$. I tried to do $z^i=e^{i\ln z}=e^{i(\ln r+i\theta+2k\pi i)}$ if $z=re^{i\theta}$, ...
2
votes
0answers
36 views

Are there famous complex constants?

Are there any famous constants (like $\pi$ and $e$) that are complex? More specifically, to rule out trivial complex numbers, are there any famous constants of the form $a+bi$ with $a \neq 0 \wedge b ...
0
votes
1answer
20 views

Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ ...
2
votes
1answer
30 views

Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
-1
votes
1answer
24 views

Cartesian equation of the loci of 'z', [on hold]

Can someone FIND |z-3j|+|z+7j|=12
1
vote
1answer
31 views

Consider the equation $|z + 3i|=3|z|$ for complex z and give a geometric description of the set S of all solutions.

Writing $z$ in the form $a+ib$ and then rearranging gives $-8a^2-8b^2+6b+9=0$. The most promising form I could manage from this is $(b-\frac{3}{8})^2=(\frac{9}{8}-a)(\frac{9}{8}+a)$ but I still do not ...
-9
votes
1answer
53 views

Can someone find z^10 quickly, given z? [on hold]

z = 1 + i z^10 = ? I need a solve of this quick please !
0
votes
1answer
28 views

How can I split this into its' real and imaginary parts, and simplify?

Essentially, I want to prove that $| \sum_{k=1}^n e^{ik}|$ is bounded. If I obtain an expression for this sum: $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ I am not sure how to proceed ...
0
votes
1answer
43 views

Help with simplifying this - where have I made an error

$$e^i \frac{e^{in}-1}{e^i-1}$$ $$e^i \frac{e^{in}-1}{e^i-1} \cdot \frac{e^{-i} -1}{e^{-i} -1}$$ $$e^i \frac{(e^{in}-1)(e^{-i}-1)}{(e^{-i}-1)(e^i-1)}$$ $$e^i ...
3
votes
5answers
44 views

Find the root of a complex number

Find all complex numbers $z$ such that $$z^2=12−16i,$$ and give your answer in the form $a+bi$. We set $$z= a+bi,$$ thus, $$z^2 = (a^2 - b^2) + (2ab)i.$$ Equating both $z^2$ we have $$ a^2 ...
2
votes
1answer
30 views

How to solve a Complex equation involving the conjugate: $(x+iy)^2-2(x-iy)+1=0$

I want to find a Complex value for $z$ that satisfy the equation: $$z^2-2z^*+1=0$$ But i have never seen the conjugate taking part of an equation. What i have tried is give $z$ some components ...
0
votes
2answers
23 views

How do you compute an expression containing complex numbers with large powers?

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$ $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$ So I have these equations with the answers ...
0
votes
1answer
27 views

Solve z in an expression involving complex conjugates.

Solve for z, and give your answer in the form a+bi. $$\overline{z+2-2i} = {2z + 5 - 7i}$$ I know fully understand the concept of complex numbers and complex conjugates. I've found that the answer is ...