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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1
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4answers
41 views

finding real roots by way of complex

I was given $$x^4 + 1$$ and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached. My teacher first found 4 ...
-1
votes
0answers
33 views

How to calculate with complex arguments?

I've got problems with calculating the complex argument of a number. We defined it like this: Let $$f: \Bbb C \ -{0} \to \Bbb [0;2π)$$ for $x=0$ we define no argument. For $zw≠0$ it is: $$Arg(zw)= ...
1
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1answer
26 views

Having trouble understanding(The proof of); $A \subset \mathbb{C}$ is finite implies the limit point set of $A$ is empty.

So I am self-studying Complex Analysis. The idea behind the proof is clear, and I understand the intuition etc. There are a few statements that are really giving me trouble accepting this proof ...
0
votes
0answers
14 views

Is this sum differentiable?

Let $R$ be an infinite set of complex numbers $\rho$ with $0<\Re(\rho)<1$, $x$ be a nonzero real number and consider the summation $\sum_{\rho} x^{\rho}$. Is this sum differentiable with respect ...
-3
votes
1answer
41 views

Proving a fact about complex numbers [duplicate]

How to prove the following: For any complex numbers $z_1,...,z_n$, there exists a subset $E$ of $\{1,...,n\}$ s.t. $$\left| {\sum\limits_{j \in E} {{z_j}} } \right| \geqslant ...
3
votes
1answer
23 views

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$?

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$? Let $z=r(\cos(\theta)+i \sin(\theta))$. Then $\frac{\Re(z)}{|z|} = \frac{r \cos(\theta)}{r} = \cos(\theta) $; as the ...
0
votes
2answers
13 views

Manipulating exponents in complex numbers

I have had this problem for a long time now: Suppose we take the cube root of unity $\omega$. $\omega ^ 2 = e ^ {i {4\pi\over3}} = {(e ^ {i 4\pi})}^{1\over3} = {(e ^ {i 2\pi})}^{1\over3} = e ^ {i ...
0
votes
1answer
29 views

Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly continuous on $(-1,1)$

Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly continuous on $(-1,1)$. Partial proof : Suppose $f$ is uniformly continuous. $\implies \forall \epsilon > 0, \exists \delta ...
0
votes
0answers
21 views

Polar Equations (Complex)

So I'm trying to figure out what the angle $θ$ would equal at $x=-2$ for the polar equation $r=θ+sin(2θ)$. All I know is that $θ$ has a domain of $0\leθ\le \pi$ and $y < 1$ (pretty sure). I ...
2
votes
1answer
27 views

Why is this not a complex variable?

I'm watching a video about the Jacobi Theta Function on YouTube. At around 6:14 in the video, he has shown that... $$\vartheta (x) = \sum_{n\in \mathbb Z} e^{-\pi {n^2} x} = \sum_{k\in \mathbb Z} ...
-1
votes
1answer
55 views

How to solve this ${5z+5i}\over 2z^2+2$ complex number [on hold]

I've tried solving this but i keep getting my answer wrong? I made the bottom all real numbers and multiplied it with the top as well. why is my answer wrong
0
votes
1answer
15 views

Show that the sets of isolated point of $E$ is a countable set - Axiome of choice

Let $E \subset \mathbb{C}$. Show that the sets of isolated point of $E$ is a countable set. That question is related to this question. However, my question somewhat different. Define ...
1
vote
3answers
37 views

If $|z-i|<1$ what can we deduce about $|z-1|$ and $|z+i|$

If $|z-i|<1$ what can we deduce about how large or small $|z-1|$ and $|z+i|$ could be? I tried drawing a diagram to get a feel but I don't know how to do anything more. I feel like $ 1< ...
3
votes
0answers
19 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
0
votes
0answers
44 views

How to find branch points for complex functions?

I'm looking for a standard way I can approach problems where I am tasked to find the branch points and branch cuts of a complex function. For instance, $$ f(z) = e^{(z^2+1)^{1/2}}$$ or $$ f(z) = ...
0
votes
0answers
45 views

Is there a name for the two parts of a complex number?

A complex number is the sum of a real number and an imaginary number. Is there a collective name for the two parts comprising a complex number, such that when used, it is (pretty) clear that the ...
0
votes
1answer
26 views

Parametrize the given curve and compute the integral (complex numbers)

The integral I have to evaluate is $\int_Czdz$, where $C$ is the line from 0 to $1+i$, and then from $1+i$ to 2. My work: $z_1(t)=(1+i)t$ and $z_2(t)=(t+1)+i(1-t)=t(i-1)+(1+i)$, $t\in[0,1]$. ...
7
votes
3answers
155 views

How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
2
votes
0answers
20 views

get magnitude of addition of complex numbers in trigonometric form

My problem is that I have multiple complex number in trigonometric form and I want to add those and get the magnitude of the result. I am aware that the normal route would be to calculate the ...
-3
votes
0answers
30 views

Does the Riemann Hypothesis consider mirror symmetry on its non-trivial zeros?

Setting the bottom corners of the square 1 on the center of two intersected circumferences and taking as center of symmetry the center of that intersection, it's possible to project the square 1 ...
3
votes
1answer
23 views

If $z_0$ is a root of the equation $z^n\cos\theta_0+z^{n-1}\cos\theta_1+\cdots+\cos\theta_n=2$

If $z_0$ is a root of the equation $z^n\cos\theta_0+z^{n-1}\cos\theta_1+\cdots+\cos\theta_n=2$, then $|z_0|<1/2$ $|z_0|>1/2$ $|z_0|=1/2$ Using triangle ...
1
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0answers
46 views

Find a Mobius Transformation that carries the points $ -1, i, 1+i$ to the following:

My goal is to find a Mobius transformation that transforms $-1, i, 1+i$ onto the points a) $0, 2i, 1-i$ b) $i, \infty, 1$ For part a, I know that the Mobius transformation $M$ will be such that ...
7
votes
1answer
101 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
10
votes
4answers
173 views

Why isn't $e^{2\pi xi}=1$ true for all $x$?

We know that $$e^{\pi i}+1=0$$and $$e^{\pi i}=-1$$ So$$(e^{\pi i})^2=(-1)^2$$$$e^{2\pi i}=1$$ Because $1$ is the multiplicative identity,$$(e^{2\pi i})^x=1^x$$$$e^{2\pi xi} =1$$should also hold ...
1
vote
5answers
79 views

What is the square root of $i$?

I started by assuming it is also a complex number but I'm finding it impossible to see the correct way to do it. If we say $i = (a + bi)(a + bi)$ then $a^2 - b^2 + 2abi = i$ equating real and ...
3
votes
2answers
36 views

How to justify $a=(a,0)$ in Theorem $\mathbf{1.29}$ in Baby Rudin?

Rudin says in page fourtheen in theorem 1.29 : If $a$ and $b$ are real, then $(a,b)=a+bi$. Proof he gives: $$a+bi=(a,0)+(b,0)(0,1)\\=(a,0)+(0,b)=(a,b)$$ of course this is correct (if we accept ...
1
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2answers
60 views

What is the $\sqrt{-1}$ when working in a quaternion space?

I dont think I really need to elaborate, do I? If you know what quaternions are then you know there are several imaginary-value options to choose from, or axes, along which the $\sqrt{-1}$ may exist. ...
0
votes
0answers
34 views

Show that $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a complex number)

How do I prove this? Suppose that $a, b$ and $c$ belong to $\mathbb C$ and that $$\lim_{z\to z_0} f(z)=a$$ and $$\lim_{z\to z_0} g(z)=b.$$ a - $\lim_{z\to z_0} cf(z)=ac$ (where $c$ is a ...
2
votes
1answer
38 views

Finding roots of complex number

The problem is specific as an example from hw. But it is more the concept/process I could use clarification on. Given a complex number $$\Big(\frac{-2}{1-i\sqrt3}\Big)^{\frac{1}{4}}$$ Find all ...
1
vote
1answer
88 views

Solve $e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0$.

Solve $$e^{4z} +e^{3z} + e^{2z} + e^z + 1 = 0.$$ I have attempted this problem with the usual definition by writing $z=x+iy$ and using $e^z = e^x(\cos y + i \sin y)$ but got a large unsolvable mess. ...
2
votes
1answer
38 views

Are there complex numbers whose sines are zero?

I recently learned that $\sin(z)$ has an extension into the complex plane, namely: $$\frac{e^{iz}-e^{-iz}}{2i}$$ Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ? I am ...
3
votes
1answer
56 views

Show that the elements of the form $1+\zeta + \zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$

Let $\zeta = e^\frac{2 \pi i}{p}$, with $p$ prime. Show that the elements of the form $1+\zeta +\zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$. I know ...
0
votes
1answer
39 views

How to bound this complex number from below?

I am doing an $\epsilon-\delta$ proof ($z \rightarrow i, f(z) \rightarrow \infty$) and currently have the absolute value $$|f(z)|=\left|\frac{z-1}{z^2+1}\right|$$ and I wish to make a statement about ...
1
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1answer
18 views

Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
0
votes
1answer
27 views

Complex Numbers in Factoring [closed]

Why does "$i$" only get involved in factoring a function when there is a ($+$) in the equation? EX: $x^2 + 9$.
1
vote
1answer
17 views

How to express an angle in terms of pi

I have the complex number $z = 5 + 6i$ in polar form $$z = \sqrt{61} (\cos \theta + i\sin \theta)$$ and $$\theta = \tan^{-1}\left(\frac{6}{5}\right) = 0.87605805059 \text{ rad}$$ But I need that ...
1
vote
0answers
28 views

Find maximum value or upper bound of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$ [duplicate]

If $|z_{1}|=2,|z_{2}|=3,|z_{3}|=4$,then find maximum value of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$. My attempt:I considered 3 circles having centre origin and radii as $2,3,4$. Then I ...
-1
votes
1answer
99 views

$z^n=(i+z)^n$, solve for $z$

I came across this question from an older textbook with no answers and I'm a bit stuck. Currently, I have done the following; let $z=r(\cos x+i \sin x)$ ($z \in \mathbb{C}$) therefore, we now have ...
0
votes
2answers
42 views

Continuity of a function with complex variables

How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$? $$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i ...
2
votes
1answer
38 views

Calculate complex eigenvector

Hi i have problem i hope that someone can make this for me more clear: So i have matrix $A = \begin{bmatrix} -2 & 1 \\ -2 & 0 \\ \end{bmatrix}$ I have to calculate eigenvector as matrix $P$ ...
2
votes
3answers
36 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
0
votes
1answer
24 views

Find the real and imaginary parts in the given expression:

$$(z+1)^2=u(r,\theta)+iv(r,\theta)$$ We are learning how to apply the polar form of the Cauchy-Riemann equations. I understand how to do this using the C-R equations the non-polar way, but I'm at a ...
2
votes
1answer
27 views

How do I express this $f(x,y)$ in terms of $f(z)$

$$f(x,y)=e^y\sin x+ie^y\cos x$$ The problem requires me to express in terms of $z$ only. My attempt: $$=e^y(\sin x+i\cos x)$$ $$=ie^y(\cos x-i\sin x)$$ If $e^{-i\theta}=(\cos\theta - i\sin ...
0
votes
0answers
12 views

Adding a constant after finding harmonic conjugate

The question asked me to find the harmonic conjugate of $u$ and then express $f(x,y)$ in terms of $z$. I found the conjugate. I also reasoned that it must be the expanded form of $z^2$. However, they ...
0
votes
2answers
20 views

Is this function harmonic?

$$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ ...
0
votes
2answers
15 views

Radius of convergence for complex power series

I am supposed to find the radius of convergence for the complex power series $$\sum_{n=0}^{\infty}(-1)^n2^nz^{2n+2}$$ I know that the radius of convergence is calculated by ...
0
votes
1answer
17 views

Help finding a second homogeneous polynomial of degree 5 that are also harmonic

Essentially I have to find 2 homogeneous polynomial of degree 5 that are also harmonic. Knowing z=(x+iy) is analytic I found my first polynomial to be f(z)=z^5 and that multiples of this would ...
12
votes
1answer
202 views

Proving that $e^\pi=e^{-\pi}$

I've been stuck with this for a while now. I have this chain of reasoning that would imply $e^{-\pi}=e^\pi$, obviously false, since $e^\pi$ and $e^{-\pi}$ are two real distinct numbers and so I must ...
2
votes
3answers
46 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...
0
votes
1answer
28 views

Simplify: $2w(z^2+1)=2z(w^2+1)$, $z,w \in \mathbb{C}$

I'm trying to simplify: $$2w(z^2+1)=2z(w^2+1), z,w \in \mathbb{C}$$ Simplifies to: $$2wz^2-2zw^2+2w-2z=0$$ But doesn't seem like it would form a nice quadratic function. Any tips? Would it be ...