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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0
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0answers
20 views

randomly generate M pairs of complex numbers from 1 to N, find gcd.

I want to write a script to generate statistics on gcd's and number of steps required to find them by the Euclidean algorithm, using M randomly generated pairs $a+bi$ between 1 and N. And plot them. ...
0
votes
1answer
21 views

When is the radius of convergence of the product of two complex power series twice the radius of convergence of the product of the radii

Proving that the product has a larger radius then the product isn't too bad using the nth root test, but another practice question I have asks for examples of power series $\sum a_kz^k$ with radius of ...
28
votes
7answers
3k views

What does it mean to divide a complex number by another complex number?

Suppose I have: $w=2+3i$ and $x=1+2i$. What does it really mean to divide $w$ by $x$? EDIT: I am sorry that I did not tell my question precisely. (What you all told me turned out to be already known ...
2
votes
0answers
46 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
0
votes
0answers
31 views

What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $$p=1+\mathtt{i}=\bar{q},\ \ q=1-\mathtt{i}=\bar{p}.$$ Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & ...
0
votes
1answer
51 views

Can I express some power of $\cos(\frac {2\pi}{5})$ as a rational number without using complex numbers?

I have been trying to express a power of $\cos(\frac {2\pi}{5})$ as a "rational number", or trying to find a "rational number" that results from a linear combination of powers of $\cos(\frac ...
0
votes
1answer
35 views

Residue of $1/(\sin(1/z))$ defined at $z=0$? Trying to derive Laurent Series of $\csc (1/z)$ to find it.

This question is related to this one. I was able to figure out on my own that the residue of $\displaystyle \sin \left(\frac{1}{z} \right)$ is defined at $z=0$ by finding the Laurent Series of ...
-1
votes
1answer
14 views

Converting from complex to sinusoidal form and vise versa [closed]

I'm having some trouble understanding this type of transformation. The materials provided by my professor doesn't even mention the method that is being used to switch from complex to sinusoidal and ...
0
votes
0answers
33 views

Cartesian equation from the complex equation

Find the Cartesian equation for the curve corresponding to the equation $|{z+8\over 16j-z}|=3$ Describe what curve is represented by the equation. Does my answer look correct? $|z+8|=3|16j-z|$ set ...
7
votes
1answer
194 views

Determining if a complex number is a root of unity

How would you determine if $a+ib$ is an nth root of unity? Obviously, the modulus of $a+ib$ must be $1$. But you would also need to determine if $a+ib$ is located at a vertex of a regular ...
1
vote
1answer
19 views

Complex conjugation of fractional powers

I would like to know when the complex conjugate can be moved outside of a real power, i.e. when is it true that $$(\overline{z})^p = \overline{z^p}$$ where $p$ is real? I wrote $z$ in exponential ...
10
votes
3answers
462 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
vote
0answers
16 views

Method for determining where Laurent series converge

I have to find the Laurent series for $f(z) = \frac{1}{z^2(4z-1)}$. I know there are two series, centered at 0 and at 1/4, because that is where $f$ is not analytic, and I found the series using the ...
0
votes
4answers
57 views

Prove that a product of two complex numbers has zero imaginary part

This is my homework, which reads as follows: Let $z_1, z_2$ be complex numbers. Prove that when $z_1z_2 \neq -1$ and $|z_1| = |z_2| = 1$, then the imaginary part of $$ \frac{z_1 + z_2}{1 + z_1z_2} $$ ...
1
vote
1answer
24 views

Does every non-trivial $\mathbb{C}$-algebra contain an element which is not a square?

Let $A$ be a $\mathbb{C}$-algebra. Letting $i$ be the imaginary unit, then for each $a \in A$ we have $$ a = \left(\frac{a-1}{2}\right)^2 + \left(i\frac{a+1}{2}\right)^2 $$ such that each element of ...
2
votes
0answers
26 views

Graphing/visualizing a complex parametric plot without using mathematica

I am trying to visualize the parametric plot in $\mathbb{C}$ of the curve $\gamma$ defined for $t\in[-\infty,\infty]$ as $$\gamma(t)=\exp\left(-t^{2}+\frac{t}{\sqrt{1+t^2}}i\right).$$ I think I find ...
2
votes
2answers
162 views

What is $\lfloor i\rfloor$?

So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. ...
0
votes
2answers
15 views

Definition of complex argument: E.g. what is the locus $\arg z \geq 3 \pi/4$?

Say that $\arg z$ has principal values $(-\pi,\pi]$. Then should the locus of points $z$ such that $\arg z \geq 3 \pi/4$ be understood to mean ... Just the region bounded by the rays $\arg z = 3 ...
0
votes
2answers
36 views

Among complex $z$ such that $|z-25i|\leq 15$, which have…

Among the complex numbers $z$ which satisfies $|z-25i|\leq 15$, find the complex number $z$ having: (A) Least positive argument (B) Greatest positive argument (C) Least modulus (D) Greatest ...
0
votes
1answer
55 views

Primitive of $\frac{1}{z}$ [closed]

I want show that the function $w: \mathbb{C}^{*} \rightarrow \mathbb{C}$ determine by: $$ w(z) = \frac{1}{z} $$ Hasn't a primitive function defined in $\mathbb{C}^{*}$, I have a primitive function of ...
0
votes
1answer
34 views

Fractional Exponents Confusion

Let a and b be natural numbers (not including zero). Is it true that will not equal for all possible solutions? For instance, if a=b the would always give an output of x (assuming you don't start ...
0
votes
1answer
28 views

How to change complex numbers into polar form? [closed]

How do I changecomplex numbers, for example $2+3i$ to polar form of $re^{i\theta}$. Thank you for any answers.
1
vote
2answers
22 views

Laurent Series about $z=0$ of $f(z) = \frac{1}{z^3 - iz}$

So far: $$ \frac{1}{z^3 - iz} = \frac{1}{z(z^2 - i)} = \frac{i}{z} - \frac{iz}{z^2 - i} $$ Now I see that: $$ \frac{-iz}{z^2 - i} = z\left(\frac{i}{i - z^2}\right), $$ and this is where I get stuck. ...
2
votes
1answer
71 views

How can I express $ i^{2i}$ in the form $x + iy$?

I'm not sure how to begin since this is not in the form $re^{i \theta}$.
2
votes
1answer
70 views

If $|z_n-z_m|> 2$ for every $n\ne m$ then $\sum \frac{1}{z_n^3}$ converges

Let $(z_n)$ be a sequence of non-zero complex numbers such that $\forall n,m, n\neq m\implies |z_n-z_m|> 2$ Prove that $\sum \frac{1}{z_n^3}$ converges. I'm clueless with this problem. A ...
0
votes
1answer
35 views

How to find the derivative of $f(z)$ if $z\in \mathbb C$

How to find the derivative of $f(z)$ if $z\in \mathbb C$ Let $z=x+iy$, then $f(z)=u(x,y)=iv(x,y)$ is it simply $u_x+iv_x=u_y+iv_y$?
0
votes
2answers
51 views

Quick Question - Complex Roots of Polynomials?

I'm asked to solve for Z where $$\frac{z+i}{2z-i} = \frac{-1}{2} + i\frac{\sqrt 3}{2}$$ As a result i got $$2z = \sqrt{3}zi + \frac{i}{2} - i^2\frac{\sqrt 3}{2} - i$$ The answer is supposed to be ...
-1
votes
1answer
51 views

Inequality of complex numbers involving modules [duplicate]

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$ My try: I wrote $|z|^2$ as $z\times \bar z$, but I didn't get to any result. Can ...
0
votes
2answers
24 views

Finding Possible Meromorphic functions on $\mathbb{C}$

I am trying to find all meromorphic functions on $\mathbb{C}$ such that: $$ \mid f(z) \mid \leq (\frac{3 \mid z \mid}{\mid z + 1 \mid})^{3/2}$$ Can I express the functions as: $$f(z) = ...
2
votes
0answers
42 views

Multiplying two radicals with negatives, simple algebra? [duplicate]

Evaluate $$ \sqrt{-9}\sqrt{-4} $$ Now, I am told that $\sqrt{a}\sqrt{b}=\sqrt{ab}$, so I should be able to simply write $$ \sqrt{-9}\sqrt{-4} = \sqrt{(-9)(-4)}=\sqrt{36} = 6 $$ However, I am also told ...
2
votes
1answer
50 views

Show that $\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}…\cot \frac{(m-1)\pi}{2m}=1$

Prove: $$\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}...\cot \frac{(m-1)\pi}{2m}=1$$ This is a roots of unity problem. I managed to show a similar example for $\cos$ by the ...
0
votes
0answers
18 views

Prove that the point will go 3 times around ellipse

I'd like to prove that if a point $z$ goes once around ellipse with focus $2,-2$ then point $z^3-3z$ goes 3 times around some ellipse with the same focus. I was thinking (since ellipse is a set of ...
0
votes
0answers
12 views

How can I separate the real and imaginary parts of this Ikeda mapping?

How might I separate the real and imaginary parts of this mapping? So I can plot and compare real curves. $E_{n+1} = A+BE_ne^{i\left|E_n\right|^2}$ where $E_n = x_n+iy_n$.
0
votes
3answers
50 views

Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
2
votes
1answer
28 views

Quick Question - Complex roots of polynomials?

I was asked to find solutions to $z^3 = 1$ and give my answer in Cartesian form. I got $1, -1/2 \pm i\sqrt{3}/2$ (b) Hence solve the equation $(z+i)^3 = (2z-i)^3$ Little help on this one? Any help ...
0
votes
0answers
12 views

Complex Number Powers of Coprime Rational Powers

I'm trying to figure out $z^{p/q}$ where $p,q$ are coprime. Suppose I want to find $z^{2/7}$ where $z=128$. I can rewrite $z=128e^{0}$ Now I know that the $z^{1/7}$ roots are $2e^{k2\pi i/7}$ for ...
-1
votes
1answer
40 views

Algebra Roots (Cubic/Complex)

Show that the equation $3z^3+(2-3ai)z^2+(6+2bi)z+4=0$ (where both $a$ and $b$ are real numbers) has exactly one real root, and find this root. I've dealt with quadratics in this form but never with ...
1
vote
1answer
3k views

Straight Line Equation in Complex Plane

I'm confused about the straight line equation in complex plane: how does $0 = Re((m+i)z + b)$ come from $y = mx + b$? I mean when I see $y = mx + b$, I can draw a graph in my mind, but when I see $0 ...
0
votes
1answer
76 views

How to express $\sin \sqrt{a-ib} \sin \sqrt{a+ib}$ without imaginary unit?

I got this kind of expression as a value of an infinite product: $$\prod_{k=1}^{\infty} \left(1-\frac{A}{k^2}+\frac{B}{k^4} \right)$$ It's easy to see how it can be factored into a product of two ...
0
votes
0answers
18 views

limits and convergence of sequences complex

For the following sequence discuss its limits and whether the convergence is uniform, in the region $\alpha \leq \left | z \right |\leq \beta $, for finite $\alpha$,$\beta >0$. $$\left \{ ...
0
votes
0answers
13 views

little agebra help, complex numbers

Can someone please explain this to me, I dont understand how to go from $ [ \psi-1+r( e^{2i\omega} - 4e^{-\omega}+6-4e^{-i\omega} + e^{-2i\omega})] A\psi6{n}e^{i\omega j} $ to this line here. $ ...
1
vote
1answer
40 views

Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$

Specifically, $\displaystyle f(z) = \sin \left( \frac{1}{\displaystyle \cos \frac{1}{z}} \right)$ has singular points at $z = \displaystyle \frac{2}{\pi + 2\pi k}$, among others. Now, I am trying to ...
0
votes
2answers
54 views

Complex numbers ; $(1+i)^n=(1-i)^n$, Find $n$. [closed]

Find $n$, if $$(1+i)^n=(1-i)^n.$$ I don't really remember complex numbers, but the problem is very easy I think.
0
votes
1answer
18 views

Discrepancy between text's answer and mine: singular points of $\cot\left(\frac{1}{z}\right) - \frac{1}{z}$

The points $\frac{1}{k\pi}$, where $k \in \mathbb{Z}$ are all singularities of the function $f(z) = \cot\left(\frac{1}{z} \right) - \frac{1}{z}$. My textbook seems to think that they are simple ...
0
votes
1answer
42 views
-4
votes
2answers
88 views

Trying to derive a contradiction with this simple inequality, [closed]

I am stuck at $$(a+d)^2 - 4(ad-bc) < 0$$ $$\implies (a+d)^2<4(ad-bc)$$ $$\implies (a+d)<2\sqrt{ad-bc}$$ where $a,b,c,d \ge 0$. Is there a contradiction to derive here? Also, the square ...
3
votes
1answer
58 views

Singular point of $f(z)$ also a singular point of $1/f(z)$ and $f^{2}(z)$

Suppose $z_{0} \in \mathbb{C}$ is an isolated singular point of the function $f$ of a given type (removable, pole of order $N$, essential). I need to show that $z_{0}$ is an isolated singular point of ...
1
vote
0answers
25 views

Isolated Singular Points

I would like to check and see if my reasoning for this question is correct: Find the singular points of the function, and classify them if they are isolated singular points. Also, evaluate if ...
1
vote
1answer
15 views

How do the coefficients in the linear combination of cosines impact the number of local minima of the sum?

Consider the following function: $$f(\theta) = r_0 + r_1 \cos(\theta + \phi_1) + r_2 \cos(2\theta + \phi_2)$$ where $\theta$ is an angle between 0 and $2\pi$. For all $0\leq k\leq 2$ we have $r_k\geq ...
4
votes
4answers
155 views

How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...