Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
5
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0answers
57 views
Simplest examples of real world situations that can be elegantly represented with complex numbers
Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
1
vote
2answers
29 views
How to write in polar form
To write in polar form you use this formula
$$z=a+bi=r \left(\cos \theta+i\sin\theta \right)$$
I want the polarform for this rectangular function$$4\sqrt2(-1+i)$$
See this for more information ...
-2
votes
2answers
65 views
prove an equation of complex numbers
How to prove this equation:
$$\sin\left(\frac{\pi}{n}\right)\cdot \sin\left(\frac{2\pi}{n}\right) \cdots \sin\left(\frac{(n-1)\pi}{n}\right)=\frac{2n}{2^n}$$
There's a hint: Consider the product of ...
2
votes
3answers
78 views
Show that $z^2=2i$ iff $z=\pm(1+i)$
I am reading Beardon's Algebra and Geometry.
Show that $z^2=2i$ iff $z=\pm(1+i)$.
For the problem in question, first I made the multiplication $(1+i)\times(1+i)$ which showed the result but I ...
0
votes
2answers
46 views
Real and Imaginary Parts of $\frac{\cos(z)}{(1-e^{ix})}$
Find
$$\mathrm{Re}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$
and
$$\mathrm{Im}\bigg(\frac{\cos(x+iy)}{(1-e^{ix})}\bigg)$$
Please help I've been trying for some time now...
1
vote
1answer
33 views
A method for solving cubic equation
So I'm reading Beardon's Algebra and Geometry, and in chapter on complex numbers, author gives the following method for solving cubic equation:
Suppose we want to solve cubic equation $p_1(z)=0$, ...
0
votes
2answers
76 views
What is the principal 12th root of one?
Let $w$ be the principal 12th root of 1. What is $w$, and what are the real and complex parts of the following:
$w w^∗$ (* = complex conjugate)
$w^9$
2
votes
2answers
21 views
Small inequality on unit open disc
For $|u|,|z|<1$, $u,z$ complex numbers, how to show the inequality:
$|\frac{u-z}{1-\bar uz}|<1$?
1
vote
2answers
26 views
nasty exponentials
While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like
$$\large \rm ...
1
vote
2answers
48 views
determining residue for the purposes of calculating an integral
Determine the integral
$$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$
using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications.
In order to do this. We ...
1
vote
1answer
30 views
similarity : $z'=(1-i)z+1+i$ with the curve of $e^x-1-x$.
Let $S$ be the similarity defined by : $S(z)=(1-i)z+1+i$, for a complex number $z$ in the complex plane.
What is the image of the curve : $y=e^x-x-1$ by the similarity $S$.
My work : Let $z=x+iy$ ...
0
votes
1answer
20 views
convergence of complex series
Set that Re $z_n>=0$,$\forall$ n $\in$ N,Proof that if $\sum z_n$ and $\sum {z_n}^2$ are both convergent,then $\sum |z_n|^2$ is also convergent.
Well I've no idea how to tackle it.
1
vote
2answers
35 views
Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist?
Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist? If yes, what is its value?
0
votes
2answers
79 views
Simplification of product of complex numbers
I look for a closed formula to the expression
$$\prod_{k=1}^{n-1}\left(e^{\frac{2ik\pi}{n}}-1\right)$$
Any suggestion is welcome. Thanks.
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votes
0answers
63 views
{University Complex Analysis] contour and Laurant series [closed]
I am really lost on these problems. Please help.
$(1)$ Evaluate $$\int_\Gamma \bar z^2 dz$$ where $\Gamma$ is the following contour from $z=0$ to $z=1+i$.
$(a)$ A simple line segment
$(b)$ The ...
-1
votes
0answers
38 views
question about complex analysis [closed]
Sketch the lines defined by the following equations:
$(a)$ $\text{Re}(z^2) = r$,
$(b)$ $|z^2-1| = r$,
$(c)$ $|z + 1| + |z - 1| = r$,
where $r > 0$ is some positive, real number.
-3
votes
1answer
73 views
$\mathbb{Z}[\sqrt{-23}]$: A uniquely written set?
I suspect that
$\mathbb{Z}[\sqrt{-23}] \implies \forall~z=\sqrt{23b+a}~e^{i\arctan{\frac{23b}{a}}},~\text{where $z$ is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
3
votes
4answers
115 views
When does $az + b\bar{z} + c = 0$ represent a line?
$a,b,c$ and $z$ are all complex numbers. My idea was to show that it passes through the point $\infty$ in the extended complex plane, but I'm not quite sure how to execute that.
Update:
It says in ...
1
vote
1answer
35 views
Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]
Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows:
$$ w = a+bi \\ z = c+di $$
Can we analogically write
$$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
1
vote
2answers
72 views
How do you solve $z^4 = 2(1+i\sqrt{3})$
Solve $z^4 = 2(1+i\sqrt{3})$ in the form $r(\cos\alpha+i\sin\alpha)$ where $r>0$ and $0\le\alpha<2\pi$
I know you have to find $\arctan(\frac{\sqrt{3}}{1})=\frac{\pi}{3}$ and that is $\alpha$? ...
2
votes
2answers
50 views
Axis of glide reflection
Need to show that if $f$ is a glide reflection then there is only one line $L$
such that $f(L) = L$
What I know is that a glide reflection is an isometry
$$f(z)=a\bar{z}+b,$$ such that $|a|=1$ and ...
1
vote
2answers
24 views
Rotation of angle $\frac{\pi}{4}$ about the point $i$
Need to find an isometry which would rotate about the point $i$ by $\frac{\pi}{4}$.
So I was thinking that first I return the given point to orign, make the rotation and then translate back, right?
...
1
vote
1answer
57 views
A rather ugly limit [duplicate]
Evaluate $$\lim_{n \rightarrow \infty} n \sin (2\pi e n!)$$. I wanna ask what's wrong with my method: Define $C_n= n \cos (2\pi e n!)$ and $S_n=n \sin (2\pi e n!)$, then $C_n+iS_n=ne^{i2\pi ...
0
votes
3answers
48 views
Problem on Complex Numbers
Which of the following is most correct for the complex numbers Z and W, marked with "x" in the picture of the complex numbers below? (the dashed circle represents the unit circle)
a) $Z = W + 3i$
b) ...
1
vote
1answer
41 views
What is the modulus of a number?
What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb ...
6
votes
3answers
174 views
4 dimensional numbers
I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
1
vote
0answers
50 views
Does it make sense to talk about $ O(z)$ if $z$ complex?
Does it make sense to talk about $ O(z)$ if $z$ complex? I would have thought that the usual definition wouldn't hold, since doesn't the fact that we don't have an order on $\mathbb{C}$ change things? ...
4
votes
1answer
68 views
What are the uses of split-complex numbers?
The set of Complex numbers can be used in lots of domains like geometry, vectorial calculations, solving equation with no real solution etc. But what are the uses of split-complex number that can't be ...
4
votes
3answers
75 views
Different interpretations of imaginary number
I went through a linear algebra course and I'm a bit confused..
I think I understand the geometric interpretation of imaginary numbers - multiplying by $i$ results in rotation by $90$ degrees in so ...
1
vote
1answer
21 views
Expanding complex geometric series
I'm having trouble with part $ii.$ of the following question:
$i.$ Express the following in terms of N and z:
$$\sum^N_{n=1}2^{-n}z^n$$
Expanding with geometric series:
$$\sum^N_{n=1}2^{-n}z^n ...
6
votes
2answers
57 views
Complex Number Roots
When I am solving to find the root of a complex number what exactly am I finding? Does it relate somehow to the complex plane? What would be it's geometrical representation if it has one?
9
votes
6answers
548 views
inequality involving complex exponential
Is it true that
$$|e^{ix}-e^{iy}|\leq |x-y|$$ for $x,y\in\mathbb{R}$? I can't figure it out. I tried looking at the series for exponential but it did not help.
Could someone offer a hint?
2
votes
1answer
45 views
ring isomorphism in the complex numbers
Let $f:\mathbb{C} \to \mathbb{C}$ be a ring isomorphism for which $f(x) = x$ for all $x\in \mathbb{R}$. Prove that $f$ is either the identity mapping ($\mathrm{id}:\mathbb{C} \to \mathbb{C}$) or f ...
0
votes
1answer
46 views
Taylor Series Expansion with e and sin
Show that when $z\neq0$,
(a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$
(b) ...
1
vote
1answer
17 views
Contour Integrals for positively circular contour
Find the contour integral of $\frac{1}{(z^2+1)^2}$ for the positively oriented circular contour $|z-Ri|=R$, for every positive real number $R>\frac{1}{2}$.
I don't know how to set up the ...
1
vote
4answers
64 views
What is the polar form of $ z = 1- \sin (\alpha) + i \cos (\alpha) $?
How do I change $ z = 1- \sin (\alpha) + i \cos (\alpha) $ to polar? I got $r = (2(1-\sin(\alpha))^{\frac{1}{2}} $. I have problems with the exponential part. What should I do now?
0
votes
1answer
59 views
Evaluation of an expression regarding a complex fifth root of unity
Let $\omega$ denote a complex fifth root of unity. Define
$b_k = \sum_{j=0}^4j\omega^{-kj}$
for $0\le k\le 4$.
What is the value of $\sum_{k=0}^4b_k\omega^{k}$?
0
votes
3answers
38 views
Inequality about sum of complex numbers.
Let ${\alpha_1,\alpha_2,...,\alpha_n}$ be complex numbers, prove that
$$|\alpha_1+\alpha_2+\cdots+\alpha_n|^2 \leq n(|\alpha_1|^2+|\alpha_2|^2+\cdots+|\alpha_n|^2).$$
2
votes
2answers
50 views
Proving a complex equality
Let $a$ and $c$ be complex numbers. Show there exists complex numbers $z$ s.t. $|z-a|+|z+a| = 2|c|$ if and only if $|a| \leq |c|$.
I've shown the forward direction correctly, but I don't know how to ...
0
votes
1answer
59 views
Complex number from a region
After sketching this region$$-4\sqrt{2}\le\operatorname{Re} z\le0, $$ $$\operatorname{Im} z\ge0,$$ $$|z|\ge8$$ I need the polar form and the rectangular form of the complex number that is in the ...
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vote
4answers
70 views
Complex power of a complex number: Find $x$ and $y$ in $x + yi = (a + bi)^{c+di}$
$$
x + yi = (a + bi)^{c+di}
$$
Find $x$ and $y$ in terms of $a$, $b$, $c$ and $d$.
Where, $i$ is defined as $\sqrt{-1}$ and $a$, $b$, $c$, $d$ are real numbers.
I defined two new real number ...
2
votes
2answers
50 views
Can the expression be simplified?
If $1, a_1, a_2,\ldots, a_{n-1}$ are $n$-th roots of unity, can the following expression be simplified?
$(1-a_1)(1-a_2)\cdots(1-a_{n-1})$?
3
votes
2answers
43 views
Product rule for logarithms works on any non-zero value?
The product rule for logarithms states that:
$$\log_b M + \log_b N = \log_b (M\cdot N)$$
Most sources that I've found* state that this rule only applies when $M$ and $N$ are positive. It's true that ...
-1
votes
0answers
28 views
Is the set of complex numbers an ordered field? [duplicate]
Is $\mathbb{C}$ an ordered field?
I thought it was until my friend asked me if $1+i \le 2+3i$ makes sense. Can someone clarify this for me? Thanks!
1
vote
1answer
47 views
How to calculate an imaginary number to high exponent?
How can I calculate something like $(i+1)^{33}$ or similar high exponent without the use of a calculator?
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votes
1answer
56 views
Form of periodic function involving exponential
I am trying to prove that if the function $f(z)= a_{1}e^{\lambda_{1}z} + ... + a_{n}e^{\lambda_{n}z}$ is periodic of period $T \neq 0$ with $a_{i} \neq 0$ for every $i$, then $\lambda_{i} = 2k_{i}\pi ...
1
vote
3answers
40 views
Sum of powers of $z$ with $|z| = 1$, $z \neq 1$
I am trying to prove that for $z \neq 1$ with $|z| = 1$ the sequence of partial sums $s_{n}= \sum_{k=0}^{n} z^{k}$ is bounded. I kinda understand the reason why it should happen but I do not see the ...
0
votes
2answers
44 views
Two types of solution to the differential equation
I have read that we can have two solutions to the second order DE below, where $W$ and $W_p$ are constantants and $\psi$ is a function of $x$:
$$\frac{d^2\psi}{dx^2} = -(W-W_p) \psi $$
(a) If we ...
0
votes
1answer
39 views
Complex solutions to $a = (z+b)^n$
I have tried the whole afternoon trying to figure out how to approach an equation of the form $a = (z+b)^n$, more specifically the equation: $1 = (z+1)^4$. Is there a general approach to equations of ...
0
votes
0answers
25 views
Fractional linear transformations with given properties
I need a function of the form
$\displaystyle f(z):= \frac{az+b}{cz+d}, \qquad z\in\mathbb{C}-\{-\frac{d}{c}\}, \qquad ad-bc\neq0$
which carries the half-plane $\{z\in\mathbb{C}\ |\; ...
