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

learn more… | top users | synonyms

0
votes
0answers
4 views

Affine transformations in the complex

In $\mathbb C^2$ I have the following three lines: $r_1:3x-y+3=0, r_2:y=0, r_3:x-i=0$ I want to find all the affine transformations such that $f(r_1)=r_2, f(r_2)=r_3, f(r_3)=r_1$ How can I do it? ...
4
votes
3answers
58 views

Complex polynomial P with $P(n)= (-1)^n$

I want to show that there is no polynomial P with complex coefficients such that $P (n) = (−1)^n$ for all integers n.Does there exist an entire function with this property ? Thank you.
0
votes
2answers
38 views

nth roots of the polynomial $x^3 =2$

I have to find the solution of the polynomial x^3 - 2 =0. Attempt: $x^3=2$ $x^3=2.1 =2(cos2k\pi+isin2k\pi)$ and $k=0,1,2$ $x=2^{1/3}(cos(\frac{2k\pi}{3})+i sin(\frac{2k\pi}{3}))$ now we will get ...
1
vote
1answer
79 views

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
-1
votes
1answer
25 views

Can there be a numerical system in which logarithms can be expressed in terms of exponents in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
0
votes
2answers
36 views

Maximum value of $\frac{\alpha\overline{\beta}+\overline{\alpha}\beta}{|\alpha\beta|}$

Maximum value of $$\frac{\alpha\overline\beta+\overline\alpha\beta}{|\alpha\beta|}$$ is 1) 2 2) 1 3) none of the above. Considering $\alpha=x+iy$ and $\beta=m+in$ , on evaluating the expression I ...
0
votes
1answer
310 views

Solve $(x+iy)^2=8+6i$, for $x$ and $y$

Given $(x+iy)^2 = 8+6i$, find the values of $x$ and $y$. Hence find $\sqrt{8+6i}$. My question is when we solve we get $x = 3$ and $x = -3$, which give and $y = 1$ and $y = -1$ then Why $-3-i$ is ...
3
votes
1answer
64 views

Solutions of $ \tan(z) = \frac{z}{z^{2} + 1}$ in the complexes

In an exam I got this question: Show that if the equation $$ \tan(z) = \frac{z}{z^{2} + 1} $$ has $z_{0}$ as a solution, then $ \Re(z) = 0 $ or $ \Im(z) = 0 $ Writing $z$ as $x + i y$ seems too ...
6
votes
2answers
92 views

Find $\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$

Find $$\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$$ The general term is $\frac{1}{r}\sin\frac{r\pi}{3}$ Let $z=e^{i\frac{\pi}{3}}$ Then, ...
7
votes
2answers
272 views

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
0
votes
0answers
15 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. ...
3
votes
3answers
229 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $z_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the ...
3
votes
0answers
27 views

Cosine Inequality, Geometric interpretation in the complex plane

The following identity was given as an exercise in the course notes for a complex analysis course. I am able to solve it (the proof is given below), but am unsure of the geometric interpretation of ...
0
votes
1answer
13 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 ...
1
vote
0answers
32 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
2
votes
0answers
35 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
29 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
49 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
26 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
12 views

Converting from complex to sinusoidal form and vise versa [on hold]

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
28 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
191 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
18 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
406 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
13 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
55 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
23 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 ...
3
votes
3answers
125 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
13 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
49 views

Primitive of $\frac{1}{z}$ [on hold]

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
30 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
26 views

How to change complex numbers into polar form? [on hold]

How do I changecomplex numbers, for example $2+3i$ to polar form of $re^{i\theta}$. Thank you for any answers.
1
vote
2answers
21 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
68 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
67 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
33 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
50 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
46 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
22 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
41 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 ...
0
votes
0answers
17 views

Proof of an inequality in C ,(2)

Let $n\ge 2$is a integer,$z_{1},z_{2},\cdots,z_{n}$ are $n$ complex numbers Prove that $$\sum_{k=1}^{n}|1+z_{k}|+\dfrac{1}{n-1}\sum_{1\le i<j\le n}|1+z_{i}z_{j}|\ge\sum_{k=1}^{n}|z_{k}|$$ for ...
2
votes
1answer
47 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
11 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
47 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
27 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 ...