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
29 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
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0answers
14 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 $z_2(t)=(t+1)+i(1-t)=t(i-1)+(1+i)$, $t\in[0,1]$. ...
4
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0answers
38 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
17 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 ...
-2
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0answers
19 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
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1answer
22 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 ...
0
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0answers
39 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 ...
6
votes
0answers
63 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
134 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
75 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
34 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
vote
2answers
56 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
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0answers
32 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
37 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
85 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
55 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
37 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
vote
1answer
15 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
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1answer
27 views

Complex Numbers in Factoring [on hold]

Why does "$i$" only get involved in factoring a function when there is a ($+$) in the equation? EX: $x^2 + 9$.
1
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1answer
16 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
27 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
39 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
35 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
23 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
14 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
193 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
43 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
27 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 ...
1
vote
1answer
30 views

If $(z_{n}) \in \mathbb{C}$, $z_{n} \to \infty$ as $n \to \infty$, what happens to $|z_{n}|$, $Re(z_{n})$, $Im(z_{n})$, $Arg(z_{n})$?

Suppose the sequence $(z_{n}) \in \mathbb{C}$ converges to infinity as $n \to \infty$. I need to determine what this implies about $|z_{n}|$, $Re(z_{n})$, $Im(z_{n})$, $Arg(z_{n})$. I know that a ...
6
votes
1answer
52 views

Classification of homomorphisms $\mathbb Q \to \mathbb C^\times$

Are there any textbooks which discuss/classify the injective group homomorphisms from $\mathbb Q$ (under addition) into $\mathbb C \setminus \{0\}$ (under multiplication)?
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0answers
28 views

Multiply 3 - 2i by its conjugate. [on hold]

Please explain me, what is conjugate and how it is multiplied. Multiply $3 - 2i$ by its conjugate
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2answers
16 views

If $a, b\in\mathbb{C}$ and $ae^{2it}+be^{-2it}$ is real, then $ae^{2it}+be^{-2it}=a'\cos(2t)+b'\sin(2t)$.

I'm asked to show that If $a, b\in\mathbb{C}$ and $ae^{2it}+be^{-2it}$ is real, then $ae^{2it}+be^{-2it}=a'\cos(2t)+b'\sin(2t)$ for some $a',b'\in\mathbb{R}$ My work so far is as follows. ...
1
vote
1answer
17 views

Is the complex form of the Fourier series of a real function supposed to be real?

The question said to plot the $2\pi$ periodic extension of $f(x)=e^{-x/3}$, and find the complex form of the Fourier series for $f$. My work: ...
1
vote
1answer
15 views

What to do when there is only one valid value to be used in the Cauchy-Riemann equations

I just did 2 problems where the $u$ part of the C-R equation was $0$. I'll give one as an example. I'm confused as to what conclusions I can correctly arrive at. $$f(z)=Im(z)$$ So I can say that ...
3
votes
2answers
47 views

Representation of roots of unity.

How to represent solutions of $\sqrt[26]{1}$ with solutions of $\sqrt[26]{-1}$? I know that $$w_{k}=\cos\left(\frac{0+2k\pi}{26}\right)+i\sin\left(\frac{0+2k\pi}{26}\right), \; \; ...
0
votes
1answer
16 views

Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable.

$$ f(z) = e^{−x}e^{−iy}$$ I used the Cauchy Riemann equations to determine that $x=iy-\ln(i)$, but I'm not sure what I'm supposed to conclude. Could I say that the function is differentiable wherever ...
1
vote
2answers
37 views

Is it a removable singularity?

In the function: $$ f(z)=2iz\frac{(1-z^{2})^{\frac{1}{2}}}{1-2z^{2}} \qquad \qquad (z \in \mathbb{Z}) \,\, , $$ There is a singularity at the point $z=\pm \sqrt{1/2}$. Is that a removable ...
1
vote
0answers
22 views

Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ then is $f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$?

I am a little stuck here, suppose we have some function $$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$ then is $$f'(z)=\frac{\partial u}{\partial x}(x,y)+i\frac{\partial v}{\partial x}(x,y)$$ assuming $f$ is ...
4
votes
2answers
76 views

Are there any other solutions to this equation?

Consider the equation $1-t = tx^{1-2t}$ for some complex number $t$ and real $x$. Are there any other solutions to this equation besides $\Re(t) = \frac{1}{2}$ ? My attempt: The above equation can be ...
0
votes
2answers
42 views

Complex Analysis: Show that $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$ [closed]

How can I show that if $|a|<r<|b|$, then $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$, where $\gamma$ is the circle with center the origin, radius $r$, and positive orientation? ...
1
vote
2answers
53 views

Two complex numbers can be equal but why can't they are greater or lesser?

Yes we know that two complex numbers can be equal to one another , but why can't we say that a complex number is greater/lesser from another complex number ?
3
votes
1answer
35 views

Relation between $|z^x|$ and $|z|^x$

In the answers given to this question, the following relation is often used: $$\left| z^x \right| = \left| z \right|^x$$ with $z \in \mathbb{C}$, $z = \alpha + i \beta$. How to prove it? Can $x$ ...
0
votes
1answer
23 views

$z = 1/w$ transformation for parallel lines $y = x + b$

I am supposed to find the image of the family of parallel lines $ y = x + b $ under the transformation $w = \frac 1 z $. Attempt: Replace $x$ and $y$ with $\Re(z)$ and $\Im(z)$, respectively. ...