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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4
votes
1answer
69 views

Prove that this number is less than $1$,

a) Prove that, if $z$ and $w$ are complex numbers and $|w| = 1$, then $$\frac{|z-w|}{|1- \bar z w|} = 1$$ b) Prove that, if $|z|<1$, $|w|<1$, then $$\frac{|z-w|}{|1- \bar z w|} < 1$$ I ...
0
votes
1answer
33 views

Multiply two complex numbers

multiplication of two complex numbers - it's the same as multiplication of vectors. From physics i know that's result of multiplication of two vectors - it's a number. But when we multiply complex ...
1
vote
3answers
90 views

how did Cardano obtain three solutions for cubic?

So, if I am not mistaken Complex numbers were discovered after Cardano's method. But from Cardano's Method on Wikipedia, it says to get the three solutions, we should use the root of unity. In that ...
-2
votes
2answers
46 views

Basic Complex Number Questions [closed]

I have just started learning about complex numbers, so I would appreciate if any of you can show me the solutions to the following $2$ questions. Solve for $z$ and write your answer in rectangular ...
3
votes
4answers
77 views

Question about Euler's formula

I have a question about Euler's formula $$e^{ix} = \cos(x)+i\sin(x)$$ I want to show $$\sin(ax)\sin(bx) = \frac{1}{2}(\cos((a-b)x)-\cos((a+b)x))$$ and $$ \cos(ax)\cos(bx) = ...
0
votes
2answers
24 views

Find complex number $z$ if $arg(z^4i^{25})=arg(u)$ where $u=\frac{\sqrt{3}}{3}-\frac{1}{3}i$ and $|z|=6$

$\arg(u)=\frac{11\pi}{6}$ $t=(z^4i^{25})=2xy(y^2+xy-2x^2)+i(x^4-4x^2y^2+y^4)$ If $\arg(z^4i^{25})=\arg(u)$ does that mean $t=u$?
1
vote
0answers
32 views

How to deal with functions include complex number

If I have a complex number, say $e^{ix}=\cos x+i \sin x$. According to the definition of complex number, we know that the imaginary part of $e^{ix}$ is $\sin x$ and real part is $\cos x$. But if I ...
7
votes
6answers
111 views

A proper definition of $i$, the imaginary unit [duplicate]

Back when I was in high school, which was a long time ago, I recall my math teacher telling me that the definition of $i$, the imaginary unit, is $\sqrt{-1}$. Knowing little, at the time, I accepted ...
0
votes
1answer
25 views

Question concerning proving a function is not analytic?

Let $f$ be the function $f: C \rightarrow C$ defined by $$f(z) = \begin{cases} e^{-z^{-4}}, & z\neq0 \\ 0, &z=0 \end{cases}$$ Show that $$\lim_{z \rightarrow 0}\frac{f(z)}z$$ does not ...
2
votes
1answer
42 views

complex long division

For example we have $(2+7i)(4-i)=15+26i$. What I am after is some kind of long division method so that: $(2+7i)|\overline{15+26i}=x+yi$ If we guess $x=4$ we get a remainder of $7-2i$, but is there ...
3
votes
3answers
82 views

Factor $k^{4}+4k^{3}+8k^{2}+8k+4=0$ over $\mathbb C$

Any idea how to factor the polynomial $k^{4}+4k^{3}+8k^{2}+8k+4$ over $\mathbb C$? Candidates for rational roots are $\pm1, \pm2, \pm4$ but none of them satisfies $k^{4}+4k^{3}+8k^{2}+8k+4=0$.
1
vote
3answers
82 views

Is $0^i$ defined number and what is it value where i :is unitary imaginary part?

There are many examples of indeterminate cases, but I would like to know if $0^i$ is a defined number or is it also one of the indeterminate cases? If it is defined, then what is its value? Is it ...
3
votes
4answers
62 views

How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$?

How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$? I tried $$\lim_{z\to0} ...
1
vote
2answers
36 views

How this exponential function converted into cosine function including phase shift?

In my text book I have found this two line in an example: $$ y(t)=\frac{2}{1+j}e^{jt}+\frac{2}{1-j}e^{-jt}-\frac{1}{1+j2}e^{j2t}-\frac{1}{1-j2}e^{-j2t}$$ ...
2
votes
4answers
53 views

Complex number and exponent

If $z = -1+ i\sqrt{3}$ Is it possible that to prove by using induction $z^{2n}+2^n\cdot z^n+2^{2n}=0$ if $n$ is not multiple of $3$. I know other way of proving it.
1
vote
4answers
81 views

Show that the set $z$ satisfying $|z-z_0|=\rho|z-z_1|$ ($\rho \neq 1$) is a circle.

Prove that for $\rho \ge 0$, $\rho \neq 1$ and fix $z_0,z_1 \in \Bbb C$. Show that the set $z \in \Bbb C$ satisfying $|z-z_0|=\rho|z-z_1|$ is a circle. From the given condition I have reached that ...
1
vote
2answers
57 views

Evaluate $\binom{n}{1}\alpha_1+\binom{n}{2}\alpha_2+\binom{n}{3}\alpha_3+…+\binom{n}{n}\alpha_n$

If $\alpha_1,\alpha_2,.....,\alpha_n$ are the n;$n^{th}$ roots of unity then$\binom{n}{1}\alpha_1+\binom{n}{2}\alpha_2+\binom{n}{3}\alpha_3+......+\binom{n}{n}\alpha_n$ ...
1
vote
1answer
45 views

Properties of the principal square root of a complex number

I am studying the principal square root function of complex numbers. On Wikipedia they present a complex number $z$ using polar coordinates as \begin{equation} z = r \mathrm{e}^{i \varphi}, \quad r ...
0
votes
2answers
39 views

Complex equation - problem

So i have this equation: $$z^2 -iz=\left|z-i\right|,\quad z\in\mathbb{C}.$$ So i just used: $z=a+bi$ and got to this: $$a^2+(2ab-a)i=\sqrt{a^2+(b-1)^2}.$$ Now i have a problem: there is no $i$ on ...
0
votes
1answer
25 views

Complex number z satisfies both the inequality $|z-ai|=a+4$ and the inequality $|z-2|<1$

The number of integral values of $a$ for which at least one complex number z satisfies both the inequality $|z-ai|=a+4$ and the inequality $|z-2|<1$. I supposed $z=x+iy$ and put in both equations, ...
-1
votes
0answers
20 views

Calculate an upper bound for $\left|\frac{e^{i\alpha-\beta}-e^{-(i\alpha-\beta)}}{i\alpha-\beta}\right|$

Let $\alpha,\beta\in\mathbb R$. Calculate an upper bound for $$\left|\frac{e^{i\alpha-\beta}-e^{-(i\alpha-\beta)}}{i\alpha-\beta}\right|$$ I think that $\cosh$ is involved in the answer, but I can't ...
0
votes
1answer
34 views

Isolated singularities: removable vs poles

I understand what the singularities are, but I am having trouble establishing them in what I feel is a formal fashion. Take these two questions I am working on. $$\frac{z^4 - 2z^2 + 1}{(z-2)^2} $$ ...
1
vote
1answer
37 views

Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
2
votes
5answers
145 views

$z^3=w^3 \implies z=w$?

I've reached this in another problem I have to solve: $z,w \in \Bbb {C}$. $z^3=w^3 \implies z=w$? I've scratched my head quite a bit, but I completely forgot how to do this, I don't know if this is ...
5
votes
2answers
180 views

Commutative binary operations on $\Bbb C$ that distribute over both multiplication and addition

Does there exist a non-trivial commutative binary operation on $\Bbb C$ that distributes over both multiplication and addition? In other words, if our operation is denoted by $\odot$, then I want the ...
0
votes
0answers
11 views

Statistical significance of deviation from a complex-valued model

I have complex-valued data. At each one of about 100 linearly spaced $x$ values, I have a corresponding measurement of a complex quantity with well-defined Gaussian uncertainties on both the real and ...
1
vote
1answer
66 views

How does one use the complex plane to solve this problem?

Given: $$a^2 + ab + b^2 = 1 + i$$ $$b^2 + bc + c^2 = -2$$ $$c^2 + ca + a^2 = 1$$ Find $$(ab + bc + ca)^2.$$ The solution says to use the complex plane. Can somebody explain to me (an average ...
2
votes
1answer
52 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
51 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
109 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
89 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
73 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 ...