Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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2
votes
1answer
53 views

Geometric series and complex numbers [duplicate]

I'm new to this site, english is not my mother tongue, and I'm just learning LaTeX. I'm basically a noob, so please be indulgent if I break any rule or habits. I'm stuck at proving the following ...
1
vote
3answers
74 views

Finding all the values of $\sqrt[3]{7-4i}$

I'm reading about De Moivre's Formula and the Roots of Unity, and one of the exercises is to find all the different values of $$ \sqrt[3]{7-4i} $$ I know that you can find the $n$th root of 1 with ...
1
vote
2answers
51 views

Finding all complex roots of this equation

So i have this equation: $z^5-4z^4+11z^3+12z^2-42z+52=0 \text{ for }z\in\Bbb{C}$ One root is: $z=1+i$ That gives us also the 2nd root. $z=1-i$ But i am stuck with how to get other 3. I thought i ...
1
vote
1answer
52 views

Projection of the XY plane.

Is there a way to project the infinite Complex plane to either the Poincare disk or the unit disk - for all values of x + iy ?
0
votes
0answers
49 views

Is $i^i$a real number or not? [duplicate]

How might we go about proving $i^i$ is not a real number? I don't know in general how to exponentiate to complex powers, I found this question in an introductory calculus course, so maybe there is an ...
1
vote
1answer
38 views

Complex Conjugation problem using the identity $|x|^2=xx^*$

Show that $$|c|^2= \frac{4k^2}{k^2 +\gamma^2}$$ given (1)$$a+b=c$$ and (2)$$ik(a-b)=-\gamma c$$ This was given in a lecture without proof, so there's probably a very simple way of proving the ...
0
votes
1answer
57 views

2x2 matrix multiplication issue

Let $$f_w(z)=z+w=\begin{bmatrix}1 & w \\ 0 & 1\end{bmatrix}z$$ where $z$ is a complex number. Shouldn't this be $w$ when $z=0$? However when I do the multiplication I get ...
11
votes
1answer
93 views

Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
0
votes
1answer
44 views

What are the real and imaginary parts of this complex propagation constant?

I am currently looking at the propagation constant $\gamma\in\mathbb{C}$, which is $$ \gamma = i\omega\sqrt{\mu\epsilon-i\,\frac{\sigma\mu}{\omega}}, $$ where $i^2 = -1$ and all other quantities are ...
0
votes
2answers
63 views

How to find the real or imaginary part of an equation involving complex numbers?

I am currently using the Debye model and need to find the real and imaginary parts of the equation. The Debye equation is $$ \epsilon_\text{r} = \epsilon_\infty + \frac{\epsilon_\text{s} - ...
2
votes
1answer
78 views

Complex integration on NON-simple closed curve

Compute the following integral with the help of Cauchy's residue theorem. $$\int_C\cot z\,dz$$where , $C:z=4e^{4i\theta}$ , $-\pi\le \theta\le\pi$ Here , singularities of are given by $\sin ...
2
votes
1answer
35 views

Are all interior points limit points in complex analysis?

The definition of limit point z for a set S in complex analysis states that there exists at least one point of the set inside the deleted neighbourhood of z.Does this imply that all interior points of ...
2
votes
1answer
40 views

Question about asymptotic behaviour of argument of complex number

Let $r\in\mathbb{R}^{+}$, $\theta\in\mathbb{R}$ and $z_{0}\in\mathbb{C}$. Does $\arg{(r\text{e}^{i\theta}+z_{0})}\longrightarrow\theta$ as $r\longrightarrow\infty$?
0
votes
2answers
110 views

$i^i$ is real number. But $\ln(i^i)=i\cdot \ln(i)=\frac{i}{2}\ln(-1)$. But $\ln(-1)$ is not defined. [closed]

$i^i$ is a real number. But, $\ln(i^i)=i\cdot\ln(i)=\frac{i}{2}\ln(-1)$. But $\ln(-1)$ is not defined. So how can $i^i$ be a real number?
1
vote
0answers
18 views

Simplifying complex functions and expressions with real results

So I integrated a real function $$ \int_{0}^{k_{max}}\frac{k^4}{(k^2 + x)^2 + y^2} $$ $$= k_{max} + \frac{1}{2y} \left(i (x + iy)^{3/2} \arctan{\left(\frac{k_{max}}{(\sqrt{(x + i y})}\right)} - i (x ...
1
vote
5answers
91 views

Complex number identity by trigonometry

Show that $\lvert e^{i\theta} - 1\rvert = 2\lvert\sin(\theta/2)\rvert$ by using the geometry of the triangle with vertices 0, 1, and the midpoint of the line joining 0 and $e^{i\theta}$. I have been ...
1
vote
0answers
34 views

Finding roots of $4$th degree conjugate reciprocal polynomial

I am developing a computer program and the following polynomial, of which I need to obtain the roots, turned up $$Ax^4 + Bx^3 + Cx^2 + \overline{B}x + \overline{A}, \quad \text{where } A, B,x \in ...
0
votes
0answers
16 views

Simplifying $\sum\limits_{n=0}^N -|a_n|^2+a_na_{n+1}^\star$

Can the sum mentioned above (where we set $N+1\equiv 0$ so that the sum is cyclic) be transformed to the form $\sum\limits_{n=0}^N -|\xi_n|^2$, where $\xi_n$ are linear combinatiosn of $a_n$?
2
votes
3answers
74 views

Product of the difference of $n$th roots of $-1$ [closed]

If $w_1,w_2,\ldots,w_n$ are the $n^{\text{th}}$ roots of $-1$, then how can we prove that by mathematical induction $$(w_2-w_1 )(w_3-w_1 )\cdots(w_n-w_1 )=\frac n{w_1}?$$
3
votes
2answers
50 views

Factorisation over $\Bbb C$ of $z^2 -10z+30$

I haven't done these questions in a long time, so I am just wondering if my approach and answer is correct. When asked to $z^2-10z+30$ over $\Bbb C$, My approach: I complete the square of the ...
1
vote
2answers
29 views

A triangle and its median in complex plane.

Let $z_1$, $z_2$, $z_3$ be vertices of the triangle $\triangle ABC$. And given that $|z_1|=|z_2|=|z_3|$. Then the median through $A$ cuts the circumcircle at which point? We need to get the answer in ...
1
vote
2answers
25 views

Raising a number in Rectangular Form

What is the value of $(-2 + 3i\sqrt3)^6$? Answer is $4096$ Convert $(-2 + 3i\sqrt3)^6$ to Polar Form. $${ (\sqrt{31} \angle 111.05)^6 }$$ I use something called De Moivre's Theorem $${z^n = r^n( ...
2
votes
2answers
45 views

Complex exponential to real

I'm not yet very good at complex number, so I would appreciate the following insight: How exactly do we arrive from $e^{\pi(1-i)}-e^{-\pi(1-i)}$ to $e^{-π}-e^π$, and why does ...
4
votes
2answers
53 views

Why are values greater than $\pi$ radians given as negative in exponential form?

Find the fifth roots of $-3+3i$ in exponential form. My answers are: $$1.335e^{3i\pi/20}$$ $$1.335e^{11i\pi/20}$$ $$1.335e^{19i\pi/20}$$ $$1.335e^{27i\pi/20}$$ $$1.335e^{35i\pi/20}$$ Wolfram ...
1
vote
1answer
31 views

Multiplying square roots of negative numbers

I am just learning more about complex numbers and a question popped up I can't figure out on my own, so I've posted it here. I already know $i^2=-1$ and $i=\sqrt{-1}$ (isn't it even true that $\pm ...
0
votes
0answers
19 views

Complex Normal Gaussian noise

I would like to create complex normal Gaussian noise with dimensions $(M,N)$ The noise should have zero mean and $var=1$. How can I do so?
2
votes
2answers
54 views

Prove that for any integer $m>1$, $\ \ (z+a)^{2m}-(z-a)^{2m}=4maz\prod_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)]$.

Prove that for any integer $m>1$, $$(z+a)^{2m}-(z-a)^{2m}=4maz\prod\limits_{k=1}^{m-1}[z^2+a^2\cot^2(k\pi/2m)].$$ This how tried to do it: Expand the two brackets on the right hand side ...
0
votes
0answers
39 views

matching the powers of the coefficients of polynomials

Hi: The result of the following question is stated (as an "it is straightforward to show that" type of result) in an econometrics paper, the link of which I can provide. But I translated into a ...
2
votes
5answers
239 views

Definitions for complex numbers

I could not find this question anywhere else. But why are addition, subtraction, division, and other operations defines they are in complex numbers? Could they defined as something else?
9
votes
9answers
193 views

How to solve $z^3 + \overline z = 0$ [duplicate]

I need to solve this: $$z^3 + \overline z = 0$$ how should I manage the 0? I know that a complex number is in this form: ...
1
vote
2answers
28 views

Real Roots of Complex Quadratic Equation - (Kasana's first example)

I recently bought H.S. Kasana's Complex Variables. It seems quite interesting, and a little harder for me than I had expected, though I should be able to get through it if I take my time. ...
1
vote
6answers
78 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
0
votes
4answers
76 views

Sum of roots of unity, proving that $1+w+w^2…+w^{n-1}=0$ [closed]

If $w$ is a unit square of rank $n$ (meaning $w^n=1$), s.t $w$ is not $1$. Prove that $1+w+w^2.....+w^{n-1}=0$. We're pretty sure that we need to use induction, its easy to prove for $n=2$ but ...
0
votes
2answers
42 views

Identity with complex numbers related to the Cauchy-Schwarz inequality

I have this equation $ a_j,b_j\in \mathbb{C} , j=1,2,...,n$ $$ \left| \sum\limits_{j=1}^n a_jb_j \right|^2 = \sum\limits_{j=1}^n |a_j|^2 \sum\limits_{j=1}^n |b_j|^2 -\sum_{1\leq i \leq j \leq n} ...
1
vote
2answers
43 views

How can I visualize the interaction of the imaginary parts of the cosine/sine functions?

So I've been trying to get a good and intuitive feel for the extension of the sine and cosine functions into complex numbers (i.e. $\cos(z)$ where $z=a+bi$), and to do so I've naturally been looking ...
2
votes
2answers
32 views

Fixed points of $\frac{1\pm \sqrt{1-|a|^2}}{\bar a}.$

Prove that $\phi_a(z)=\frac{a-z}{1-\bar az}$ , $0<|a|<1$ has exactly two fixed points ; one inside the unit disc and the other outside the unit disc. Putting $\phi_a(z)=z$ I find that there ...
4
votes
0answers
60 views

If $x_1, x_2,…,x_{10}$ are such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} \cos(x_i)$ [duplicate]

Take $x_1, x_2,...,x_{10}$ such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$ with $x_1, x_2,...,x_{10}$ on $\left[0,\frac{\pi}{2}\right]$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} ...
0
votes
4answers
57 views

How find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$?

Let $k > 1$ be a real number. How may one find all complex numbers such that: $|\,1 - z\,| < k\ (1 - |\,z\,|\, )$? ...
1
vote
0answers
33 views

Complex numbers $x$, $y$, and $z$ are collinear, show that there exist $a,b,c$ for which $ax+by+cz=0$ where $a+b+c=0$

Suppose $x$, $y$, and $z$ are collinear (complex numbers). Prove that there exist real constants $a$, $b$, $c$, not all zero, such that $ax+by+cz=0$ where $a+b+c=0$. This is how I did it: First, ...
4
votes
2answers
93 views

What is the solution(s) to $1^i$?

$1^x$ is always $1$ with real numbers, but everything gets more complicated with complex numbers. Using Eulers formula, you know that $$e^{ix}=\cos(x)+i\sin(x)$$ If you make x=2π into this you'd get ...
-1
votes
3answers
73 views

$ x+y = 1 $ and $ \frac{1}{x} + \frac{1}{y} = 1 $ Solve $ x^3 + y^3 $ [closed]

$x$, $y$ are complex numbers, $x$ and $y$ aren't $0$. $$ x + y = 1 $$ $$ \frac{1}{x} + \frac{1}{y} = 1 $$ $$ x^3 + y^3 = ? $$ Thank You!
7
votes
8answers
898 views

Most natural intro to Complex Numbers [closed]

This is a soft question but I'm willing to ask. There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most ...
0
votes
1answer
86 views

Multiplicities of zeros of the polynomial $p(z)=1+2z^4+\frac 7 {10}z^{10}$

How many zeros does $p(z)=1+2z^4+\frac{7}{10}z^{10}$ have in the unit disc ? Determine multiplicities of these zeros. Let , $f(z)=2z^4$ and $g(z)=1+\frac{7}{10}z^{10}$. Then, on $|z|=1$ , ...
2
votes
3answers
65 views

Evaluate the given limit in $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$

Let , $C_r=\{re^{i\theta}:0\le \theta \le \pi\}$ denotes the semicircle traversed clockwise. Show that $$\lim_{r\to 0}\int_{C_r}\frac{e^{iz}}{z(z^2+1)}\,dz=-\pi i$$ I can not use the Jordan's ...
0
votes
1answer
51 views

Questions on whether imaginary number is larger than $0$

The imaginary number implies $i=\sqrt{-1}$ But this this not say whether $i$ is larger or smaller or equal to $0$ So i wonder if $i$ can be larger or smaller than $0$ and if so how do we see it
5
votes
1answer
100 views

Why is $\sqrt{xy}=\sqrt{x}\sqrt{y}$ also true when $x=-1$, making it $i$?

A : If: $$\sqrt{xy}=\sqrt{x}\sqrt{y}$$ only when $x,y>0$, B : Then why can I do this: $$\sqrt{-4}=\sqrt{4\times-1}=\sqrt{4}\sqrt{-1}=2i$$ which violates A since $y<0$ C : But why can I not ...
1
vote
2answers
108 views

Show that $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ represents a circle

Suppose $|a|<1$ and $r\in (0,1)$. Show that the set of complex number $z$ satisfying $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ is a circle in complex plane. Find the centre and radius of this ...
0
votes
2answers
43 views

$f$ is an entire function satisfying the given condition . Show that the function is constant

If an entire function $f(z)$ satisfies $$|f(z)|\le \frac{1+|z|}{1+\sqrt {|z|}}$$ for all $z\in \mathbb C$ then show that $f=c$ with $|c|\le 2(\sqrt 2-1)$. First we consider a function ...
1
vote
0answers
56 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
1
vote
1answer
31 views

Find all the complex numbers that satisfy this quotient.

A certain problem that I have been working on involves the equation $$1 = \frac{1}{1-n}$$ One can see that the only real-number solution is $n=0$. As far as the original problem goes, that is ...