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

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6
votes
2answers
130 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.
-6
votes
1answer
58 views
2
votes
1answer
455 views

Complex Numbers and Primitive Roots of Unity

I'm not super familiar with primitive roots of unity and I am not quite sure how to express the following problem in algebraic form. Help is appreciated, thanks :D Let R be the set of primitive ...
0
votes
2answers
28 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} - ...
20
votes
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, ...
1
vote
1answer
16 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 $C$ ...
1
vote
1answer
56 views

Solve $z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$ [duplicate]

Solve $$z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$$ I have never dealt with equations with complex numbers in them so this is interesting; first Ill expand. $$ \implies z^3 + 5z^2- 5iz + 9z + 10 - 10i = ...
0
votes
0answers
32 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
votes
2answers
59 views

Complex numbers

In an Argand diagram, the vertices on an equilateral triangle lie on a circle with center at the origin. One of the vertices represents the complex numbers 4 + 2i. Find the complex numbers that ...
0
votes
0answers
19 views
2
votes
1answer
41 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 ...
2
votes
0answers
61 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 ...
-5
votes
1answer
45 views

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

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. This is a question on complex numbers.
0
votes
1answer
49 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 ...
3
votes
0answers
50 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 ...
2
votes
1answer
14 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: ...
0
votes
1answer
62 views

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

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
5
votes
1answer
74 views

convergence of $\prod_{n=1}^\infty (1-\frac{z}{n!})$

I want to show that $\prod_{n=1}^\infty (1-\frac{z}{n!})$ is convergent (or uniformly convergent) (z is complex) Can I use the Theorem: The infinite product $\prod_{n=1}^{\infty} (1+a_n)$ converges ...
0
votes
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 $\Im[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ ...
1
vote
0answers
16 views

How to slice a complex functions about an axis that is not though the origin and skew to the z-axis?

Graphing the real part of complex function $\frac{1}{1+z^2}$ colored according to the imaginary part yields: $$Re(\frac{1}{(1+z^2)})=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}$$ ...
0
votes
3answers
37 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 ...
1
vote
3answers
52 views

complex logarithms

Using complex logarithms, how would I solve this $$\left.\frac12i\;\text{Log}\frac{1-i(1+e^{it})}{1+i(1+e^{it})}\right|_0^{2\pi}$$ would it equal; $$ \frac12i[ ln (\sqrt2) + I arg \frac{1-2i}{1+2i} ...
2
votes
1answer
34 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 ...
0
votes
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
0answers
50 views

Complex Analysis ( Open and Closed Sets) [on hold]

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
27 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
votes
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 $\text{arg}(z)$?
0
votes
1answer
21 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 $10$ ...
1
vote
0answers
22 views

Weighted undirected graphs, complex Laplacian, complex eigenvalues & spectral clusering

I am rather puzzled and confused, I have been trying to get a clear understanding of how would spectral clustering work for an undirected weighted graph, I have used the normalized Laplacian, but I ...
-1
votes
1answer
50 views

Show that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n$ [on hold]

Can you help me with the following problem. I dont have any idea how to start. Prove that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n=1$.
0
votes
1answer
17 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and ...
0
votes
2answers
16 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
0
votes
2answers
33 views

Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
0
votes
1answer
14 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...
0
votes
1answer
8 views

Question about determining accumulation points

So far the way I have determined accumulation points of given sequences or relations has been by drawing them out. However I would like some clarification to see if my thinking is correct or not. a) ...
1
vote
1answer
30 views

Conditions To Make Complex Numbers $z_1, z_2, z_3, z_4$ Vertices of a Square

Let $z_1,z_2,z_3,z_4\in\mathbb C$ be distinct. State conditions in terms of computation of complex numbers, which make $z_1,z_2,z_3,z_4$ vertices of a square (in the counterclockwise direction). ...
0
votes
1answer
14 views

Complex number and orthogonal axis

What are the properties of complex numbers which allow us to plot the real and complex part on orthogonal axis? One thing I understand is that complex portion cannot be represented as scalar multiple ...
1
vote
3answers
56 views

A complex Analysis proof

Let $a \in \mathbb{C}$ and $\phi \in \mathbb{R}$. Prove that if $|a+1|=|1+ae^{i \phi}|$ then $ae^{i \phi} = a$ or $ae^{i \phi} = \bar{a}$. I need an idea of how to approach here please anyone.
0
votes
2answers
68 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
0
votes
1answer
27 views

Algabreic manipulation with complex numbers

How does $(iwl + \frac{1}{iwc})^2$ equal to $(wl - \frac{1}{wc})^2$? Let me clarify. In physics there is the impedance which is a complex number Z = R + iwl + 1/iwc R, w, l, and c, are ...
0
votes
1answer
37 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
0
votes
1answer
33 views

Complex analysis basics

If I z = x + yi and w = f(z), describe the image R of D in the w-plane when $$0<x<\pi/2, 0<y<\infty;w = e^{iz}$$ Wouldn't this mean that in the w-plane the argument arg(w) = $\infty$ ...
0
votes
1answer
19 views

Deriving definition of the complex logarithm

Given that: $$z = Re^{i\theta} = R(a + bi) = R\left( \cos(\theta) + i\sin(\theta) \right)$$ In its polar form. $$\log(z) = \log(R) + i\theta$$ $$|z| = \sqrt{(Ra)^2 + (Rb)^2} = R\sqrt{a^2 + b^2} ...
4
votes
4answers
86 views

Considering $ (1+i)^n - (1 - i)^n $, Complex Analysis

I have been working on problems from Complex Analysis by Ahlfors, and I got stuck in the following problem: Evaluate: $$ (1 + i)^n - (1-i)^n $$ I have just "reduced" to: $$ (1 + i)^n - (1-i)^n = ...
0
votes
2answers
54 views

Math theories in Game Theory

What are all the mathematical theories in Game Theory? I have taken Mathematical Modelling, including: application of linear systems, matrix operations, inverse of matrix, leontif input-output model, ...
0
votes
3answers
41 views

Eigenvalues of skew-symmetric matrix

Prove that all of the eigenvalues of skew-symmetric matrix are complex numbers with the real part equal to 0. Has anyone got a clue how to do it?
0
votes
2answers
28 views

Understanding complex functions in w - and z - plane

I have a difficulty understanding the basics of complex functions. My exercise looks like this: "The $z$-plane region $D$ consists of the complex numbers $z = x + yi$ that satisfy the given ...
4
votes
0answers
36 views

A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
0
votes
1answer
28 views

legitimate proof of complex arguments?

If $\operatorname{Re}(z_1)>0, \operatorname{Re}(z_2)>0$, then $\operatorname{Arg}(z_1z_2) = \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2)$ My question is if the proof i used is legitimate. ...
0
votes
3answers
22 views

Find the real and imaginary part of the following

I'm having trouble finding the real and imaginary part of $z/(z+1)$ given that z=x+iy. I tried substituting that in but its seems to get really complicated and I'm not so sure how to reduce it down. ...