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

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1answer
14 views

Sum of unitary complex numbers

Let us define: $$\varphi(x,n,t):=\frac{1}{n}\sum_{y=1}^n \left| \sum_{k=1}^n e^{2ik\pi (x-y)/n + 2i \sin(2k\pi/n) t} \right|$$ Does somebody have an idea how to prove that $$ \sup_{x=1,...,n} ...
1
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0answers
15 views

A question about complex using geometric.

Let $z_{1}$, $z_{2}$, and $z_{3}$ be three distinct complex numbers. Prove that these numbers are collinear if and only if the quotient $(z_{3}-z_{1})$ \ $(z_{2}-z_{1})$ is a real number. I have been ...
0
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4answers
37 views

Prove that $az^n+b\overline{z}^n=0$ does not have any complex solutions except for $0$

Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$? The first one seems very obvious, but ...
3
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7answers
107 views

What does $\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3} $ imply?

I'm having trouble understanding what the following equality implies. $$\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3}.$$ I suspect that this means that the points form the vertices of an ...
1
vote
1answer
26 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
0
votes
2answers
31 views

Complex differentiability of $f(z)=|z|$

Why is the absolute value function $f : \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|$ not complex differentiable at any point $z_0$ in the plane?
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1answer
25 views

Complex variables Open ball [on hold]

Let $f(z) = \frac1z$ be inversion. Given a real number $a$, let $R_a = \{z \in C : Im(z) < a\}$. Why is $f(R_a)$ an open disk, provided $a < 0$. What happens when $a \ge 0$?
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2answers
33 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
0
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1answer
57 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $a_0 , a_1 , a_2 , b_0 , b_1 , b_2 \in \mathbb{C} $ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( 1 + j b_0 ) ( 1 + j^2 b_0 ) ...
3
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2answers
30 views

Square Rooting Back To Real Dimension

As we all know, square rooting -1 (a real number) opens up the "imaginary" dimension (defined by the presence of iota). We can return from the imaginary dimension back to the real dimension by ...
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3answers
33 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
-2
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2answers
78 views

What is wrong with my proof: $-1 = 1$? [duplicate]

I have some theories about why this could by wrong but I still haven't something that convinces me. What is wrong with this proof: $ -1 = i^2 = i.i = \sqrt{-1}.\sqrt{-1} = \sqrt{(-1).(-1)}= \sqrt1 = ...
2
votes
3answers
89 views

Confused with imaginary calculus

So $i$ is the complex unit and $n \in \mathbb{N} $. $$e^{2 \pi \ n \ i} = 1$$ $$1^{2 \pi \ n \ i} = 1$$ $$(e^{2 \pi \ n \ i})^{2 \pi \ n \ i} = e^{-4\pi^2 \ n^2}$$ $$e^{-4\pi^2 \ n^2} \neq 1$$ I’m ...
4
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0answers
34 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
0
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1answer
25 views

Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
1
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1answer
21 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
2
votes
1answer
18 views

Length of a complex vector

From the definition of inner product in $\mathbb{F}^n$ $$\textbf{a}\cdot\textbf{a}=\sum\limits_{k=1}^na_{k}\overline{a_{k}}$$ Say $a_{k}=x_{k}+iy_{k}$, then ...
0
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0answers
33 views

proof of a vector identity

In an exercise I am asked to prove the following vector identity: $$\textbf{a}\cdot\textbf{b}=\frac{1}{4}\big(|\textbf{a}+\textbf{b}|^{2}-|\textbf{a}-\textbf{b}|^{2}\big)$$ Both the dimension of the ...
-1
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1answer
59 views

Complex transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ [on hold]

Under the transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ is transformed to (a) $\{z\in \mathbb C:0<\operatorname{arg}(z)<\pi\}$ (b) $\{z\in \mathbb ...
0
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1answer
46 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
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votes
2answers
32 views

Modulus of a complex expression [on hold]

Show that for a complex number $a$, $$\left\lvert\frac{z-a}{1-\bar{a}z}\right\rvert = 1$$ for $|z| =1$ and $\bar{a}z ≠ 1$. I've tried to show that if $|z-a| = |1-\bar{a}z|$ then it's true but to no ...
8
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2answers
155 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
0
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1answer
40 views

stuck with a cubic equation

I picked an equation $0= x^3 +x^2 -2x -1$ I plotted it with geogebra, to see if it had more than $1$ real root. It definitely cuts the $x$-axis $3$ times. But when I checked wolfram alpha, to see ...
5
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0answers
50 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
2
votes
1answer
21 views

Seperating points in the complex plane

Given a finite set of points say $p_1,p_2, \ldots, p_n$ in the complex plane, how do I find another point $q$ such that ray $R_i$ joining $q$ to $p_i$ are all distinct. I would be happy with any kind ...
3
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0answers
78 views

Limits of sequences connected with real and complex exponential

Let us denote $S_{n}(x)=1+\frac{x}{1 !}+\frac{x^{2}}{2!}+ ... + \frac{x^{n}}{n!}$. How could be calculated the limit $$L(x)=\lim_{n\to \infty}\frac{S_{n}(n x)}{e^{n x}}=\lim_{n\to ...
0
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1answer
45 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
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0answers
40 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
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1answer
67 views

you know root square of -1, what is the larger of the square? [on hold]

there is a square ABDC, $BD = \sqrt{-1}$ what is the value of AB=BC=DC=AD?
7
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3answers
241 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
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2answers
35 views

Polar form equations on the unit circle

If $l \in [0, 2 π)$, $k, n \in N$, proof the following equations: $$\mid{e^{i k l/n} - e^{i (k-1) l/n}}\mid = \mid e^{i l/n} - 1\mid$$ and: $$\lim_{n \to \infty} \sum_{k = 1}^n \mid e^{i k l/n} - ...
1
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1answer
17 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
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0answers
19 views

Example on Complex Number using De moivrs theorem [on hold]

Prove That ( √3+i)^14 + ( √3-i )^14 = 2^14
2
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1answer
42 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
-5
votes
1answer
47 views

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. [closed]

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. This is a question on complex numbers.
2
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0answers
62 views

Can an ordered field contain complex numbers?

I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it ...
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6answers
2k views

Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
2
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1answer
18 views

Sixth root of -64 using Euler's formula and De Moivre's theorem

I am attempting to solve: $$(-64)^{\frac{1}{6}}$$ Using the relation: $$a+bi=re^{i(\tan^{-1}(\frac{b}{a})+2\pi n)}$$ And then applying De Moivre's theorem: ...
3
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0answers
51 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
0
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1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [closed]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
0
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1answer
51 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
0
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1answer
11 views

Distribution of magnitude squared for complex Gaussian

$\def\Re{\operatorname{Re}}\def\Im{\operatorname{Im}}$ If we have a random complex variable $h_l$, with $\Re[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ and $\Im[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ ...
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3answers
38 views

Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$

Let $z=\cos\theta+i\sin\theta$. Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$ Can anyone show me how to show the equation? I can't think of how to get $\frac 12 ...
0
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0answers
34 views

Question about galois imaginary and modular arithmetic

Let $p$ be a prime of type $3\space mod \space 4$. Then there is no solution $x^2 = -1 \space mod \space p$. Therefore we can define the so-called Galois imaginary $i$. ( $i^2 = -1 \space mod \space ...
0
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1answer
30 views

Complex Analysis (Limits)

Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$ Sorry for the wrong notation, I do ...
2
votes
1answer
36 views

Representation theory of $\mathbb{Z}_k$ and complex roots of unity

Is there a natural relationship between the (characters?) of irreducible representations of $\mathbb{Z}_k$ and the $k$ complex-roots of unity? Can they be like thought of as characters of its ...
-2
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0answers
55 views

Complex Analysis ( Open and Closed Sets) [closed]

I am supposed to show that if T is a closed set of complex numbers and S is contained in T, then the modulus of S is contained in T. I know a closed set means it does not extend to infinity, S=x+iy, ...
1
vote
1answer
28 views

Complex Plane ( $\arg(z)$)

Sketch the following regions of the complex plane. For each, say whether it is open, closed, or neither, and whether it is connected. No proofs necessary. $$\left\{z \in \mathbb{C}\mid -\dfrac{\pi}{2} ...
0
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0answers
23 views

If two solutions of arg$(z)$ are in interval $−\pi<$arg$(z)≤\pi$ are both correct?

For example there is complex number $z=\sqrt3-i$ Are the answers $\frac{5}{6}\pi$ and $-\frac{\pi}{6}$ correct as $\text{arg}(z)$?
1
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1answer
25 views

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the polynomials

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the two polynomials $p(x)=5ix^4-(9+2i)x^3+7x+6-i$ and $q(x)=9x^5-x^3+7x+6$. The roots, with accurate to $10$ ...