Tagged Questions
4
votes
3answers
46 views
Prove equality in triangle inequality for complex numbers
We need to show that
$$ |z_{1}+z_{2}+\cdots+z_{n}|=|z_{1}|+|z_{2}|+\cdots+|z_{n}|$$
if and only if $z_{1},z_{2},\dots,z_{n}$ have the same argument (i.e. $z_{j}=r_{j}e^{i\theta}$ for $j=1,\dots,n$).
...
2
votes
3answers
151 views
The negative square root of $-1$ as the value of $i$
I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i = (-1)^{1/2} $.
I ...
0
votes
1answer
139 views
making the domain of $z ↦\tan(z)$ injective
Given the following:
$\sin(z)$ = ($e^i$$^z$ - $e^-$$^i$$^z$)/$2i$
$\cos(z)$ = ($e^i$$^z$ + $e^-$$^i$$^z$)/$2$
$\sin(z)\cos(w) - \cos(z)\sin(w) = \sin(z-w)$
$\sin(z) = 0$ has solution $z = kπ$ for ...
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
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.
-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.
0
votes
0answers
43 views
Which complex number cannot be written in polar form?
I'm really confused by this question. Is there such a number?
0
votes
1answer
52 views
Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.
I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
1
vote
3answers
204 views
Complex numbers $z$ such that $|z|= 1$
There are infinitely many complex numbers $z$ such that $|z|= 1$. Can anybody just explain this to me so I can get the picture.
1
vote
1answer
48 views
3 complex-variable equation
Moderator Note: This is a current contest question on Brilliant.org.
$x,y,z$ are complex numbers satisfying
$$
\begin{align}
x+y+z & =1\\
x^2+y^2+z^2 & =2\\
x^3+y^3+z^3 & =3
...
0
votes
1answer
37 views
Determinant formula and invertibility.
I am working on a problem where I need to find the determinant of
$$
\begin{bmatrix}
b & a & & \\
& b & a \\
& & & \ddots \\
& & & & ...
2
votes
2answers
75 views
Finding the number of different ordered quadruples $(a,b,c,d)$ of complex numbers
Find the number of different ordered quadruples $(a,b,c,d)$ of complex numbers such that:
$$a^2=1$$
$$b^3=1$$
$$c^4=1$$
$$d^6=1$$
$$a+b+c+d=0$$
0
votes
0answers
87 views
Two questions regarding the Euclidean topology on the unit circle in $\mathbb{C}$
Let $S^1 = \{ z \in \mathbb{C}: |z| = 1 \}$ be the unit circle, with the Euclidean topology. Furthermore, we define $q : [0,1] \to S^1 , z \mapsto e^{2 \pi i z} $ . Finally, we define (for $a<b$, ...
-1
votes
2answers
69 views
How to integrate complex exponential??
Consider
$$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$
Why do we only look at the real part? What about the imaginary part ...
0
votes
2answers
44 views
Proving two Complexes' Numbers Properties
I'm having problem working with complex number on this question and was wondering if someone can walk through with me their reasoning on how to solve this/these types of questions. Thanks in advance!
...
1
vote
2answers
123 views
Sketch the Set in Complex Plane
I don't understand how to sketch a set in the Complex Plane. So i would appreciate if someone could explain it to me. What do i have to know ? Which formulas do i need ? Can someone explain how a open ...
0
votes
1answer
52 views
Prove that one of the roots of a poly lies outside the unit circle in the complex plane
please could you help me with the following problem. I need to show that a particular multistep method in numerical analysis satisfies the root condition. I have reduced the problem to showing that ...
1
vote
1answer
54 views
$\lim_{t \rightarrow \infty}\frac{t^n z^n}{|t^nz^n + \cdots+tz +c|} $?
How to find the limit
$$\lim_{t \rightarrow \infty}\frac{t^n z^n}{|t^nz^n + \cdots+tz +c|} $$
where $c \in\Bbb C$?
is the answer $z^n$? please help :)
1
vote
2answers
66 views
Need help with matrix multiplication: $ (aI + bJ)(cI + dJ) $.
Consider the matrix
$$
A = \left[ \matrix{a & -b \\ b & a} \right],
$$
and write this as $ A = aI + bJ $, where
$$
I = \left[ \matrix{1 & 0 \\ 0 & 1} \right] \quad \text{and} \quad
J = ...
2
votes
2answers
43 views
what's the conjugate of $i^{-\frac{1}{2}}$?
If a complex number is $A=a+bi$, then its conjugate is $\bar{A}=a-bi$.
What's more, the conjugate of $e^{i\theta}$ is $e^{-i\theta}$. Well, it is known to us.
Now, if a complex number is ...
1
vote
1answer
18 views
$(y^2+2iy)^{m-\frac{1}{2}}=y^{m-\frac{1}{2}}(2i)^{m-\frac{1}{2}}+O(y^{m+\frac{1}{2}})(0\leq y \leq 1)?$
This question comes from stein's book Introduction to fourier analysis on euclidean space,page 159.
when $m > 1/2$,
...
3
votes
2answers
258 views
Circuit Analysis - series RC Circuit with AC supply
If anyone is familiar with Horowitz and Hill... its exercise 1.19
Show that all the average power delivered to the preceding circuit winds up in the resistor. Do this by computing the value of ...
4
votes
3answers
147 views
Calculate the $\int_0^{2\pi}\cos(mx)\cos(nx)dx$
I'm having trouble with this problem:
Consider the integral:
$$\tag 1\int_0^{2\pi}\cos(mx)\cos(nx)dx$$
a. Write $\cos(mx)$ and $\cos(nx)$ in terms of complex exponentials and compute ...
1
vote
1answer
60 views
Problems with basic algebra
I'm studying for an exam in a digital communications course I'm taking, and the solution to one question has me totally lost. While finding the Inverse Fourier Transform of a function, there's one ...
1
vote
1answer
99 views
Prove Using Complex Multiplication
Show the folowing general arctangent identity using complex multiplication,
$\arctan\frac{1}{a-b} = \arctan\frac{1}{a} + \arctan\frac{b}{a^2-ab+1}$,
for distinct real numbers $a$ and $b$.
0
votes
2answers
59 views
Euler's Formula Question.
Find
$\cos\theta + \cos3\theta + ... + \cos((2n+1)\theta$,
and
$\sin\theta + \sin3\theta + ... + \sin((2n+1)\theta$.
Where $\theta \in$ Reals.
1
vote
2answers
132 views
Complex number inequality.
If z and w are distinct complex numbers such that $|z| =|w| = r$, prove that
$|\frac{1}{2}(z + w)| < r$.
1
vote
1answer
149 views
Sum of a complex, finite geometric series and its identity
I have the formula for summing a finite geometric series as $$1+z+z^2\cdots +z^n = \frac{1-z^{n+1}}{1-z},$$ where $z\in\mathbb{C}$ and $n=0,1,...$. I am asked to infer the identity $$1+\cos\theta+\cos ...
3
votes
2answers
117 views
Prove by Induction
For $n\in \mathbb{N}$ and $z\in \mathbb{C}$:
$\sin{(nz)}=\sum _{ k=0 }^{ n }{ \binom{n}{k} }\frac{1}{2i}(i^k-(-i)^k)(\cos{z})^{n-k}(\sin{z})^k $
$\cos{(nz)}=\sum _{ k=0 }^{ n }{ ...
0
votes
3answers
88 views
Cartesian and Polar Coordinate
I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers:
a) $z_{1}=-i$
b) $z_{2}=\sqrt{3}+i$
...
0
votes
1answer
28 views
Need help with understanding exponential form.
I'm looking at this example in my book:
$$z = -1 - i$$
The book doesn't explain how it got to the exponential form, which is:
$$\sqrt2e^{-i3\pi/4}$$
I understand how $3\pi/4$ was found, but I don't ...
0
votes
1answer
52 views
Linear set of equations with complex numbers
How can I solve a linear set of equations with complex numbers? I haven't solved a set of equation with complex number before, so I'd like to know if there are particular rules to follow..
Thanks for ...
3
votes
3answers
92 views
The distance from 1 to the other $n$th roots of unity
I want to prove that the sum of the fourth powers of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $6n$. I consider the distance from 1 to the other $n$th roots of unity ...
2
votes
2answers
66 views
The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle
I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
0
votes
1answer
83 views
Sketching complex numbers in coordinate system
Hi guys i want to sketch these set of complex numbers in coordinate system, i hope you can help me.
$a.\{z\in \mathbb{C}||z-1|+|z+1|<4\}$
$b.\{z\in \mathbb{C}| \mathrm{Im}((1-i)z)=0\}$
...
0
votes
0answers
62 views
Prove that $x-\mid y\mid\le\left|\frac{x-y}{1-xy}\right|<1$
Let a be a real number, b is a complex number, $a \in (0,1)$ and $|b|<1$
Prove that $$x-\mid y\mid\le\left|\frac{x-y}{1-xy}\right|<1$$
I have solved the left side: ...
1
vote
2answers
100 views
show the rule between $w$, $w^2$, …$w^{n-1}$ with $w^n$ given w an nth root of unity
Let $w≠1$ be an $n$-th root of unity, i.e., $w^n-1=0$. Show that
$1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}$.
My question is how to relate $w, w^2, \dots,w^{n-1}$ with $w^n$?
0
votes
2answers
153 views
Give a geometric description of the following set
Give a geometric description of the following set:
$$\{z:|z-2|+|z+2|=5\}$$
2
votes
2answers
139 views
Geometric description of set of points satisfying $Im(z)>0$
How can we describe geometrically the set of points $z$ satisfying the condition $Im(z)>0$ where $z$ is a complex number?
2
votes
2answers
98 views
Complex Eigenvectors
This is a relatively simple question and I've Google'd this topic but I can't seem to grasp the method. One site I visited helped be a bit, but it simply used substitution to solve the problem rather ...
2
votes
0answers
88 views
complex numbers
I know that $f$ is continuous on $[a,b]$ with $ab\neq0$, $f(a)f(b)\neq0$ and the complex numbers:
$$ z = a^2 + f(a)i $$
$$ w = b^2 - f(b)i $$
$$|\bar w + z| = |w - \bar z|$$
1)Prove that $w\cdot z$ ...
1
vote
2answers
103 views
A question on complex numbers
We are given
If $\cos(a+ib)$=$r (\cos\theta +i\sin\theta)$
then prove that $e^{2b} = \sin(a-\theta)/\sin(a+\theta)$
I just tried and got $b = 0$ such that $\cos(a) = ra$. Will there be other ...
1
vote
0answers
55 views
exercise on complex numbers
Let
$$f(z)=\frac{z-a}{z-b}$$
with $a,b\in D(0,r)$ and $r>0$. Let
$$E=\{z\in\mathbb C: f(z)\notin N\}$$
$$N=\{Re(z)\leq 0;Im(z)=0\}$$
How can i find $E$ in terms of $r$?
0
votes
2answers
177 views
Contour integration that is reduced to integration over unit circle
I want to evaluate $\displaystyle \frac{1}{2\pi}\int_{0}^{2\pi} \frac{1}{1-2rcos\theta + r^2} d\theta$ for $0 < r< 1$.
I was thinking or replacing $2cos\theta = (e^{i\theta} + e^{-i\theta}) $ ...
1
vote
3answers
106 views
Non-existence of a certain holomorphic function on the unit disk
I am trying to prove the following:
Let $n\in \mathbb{N}$. Prove that $\not \exists$ a holomorphic function $f$ on the open unit disk satisfying:
$f\left(\displaystyle \frac{1}{n}\right) = 2^{-n}$
...
1
vote
1answer
36 views
proving that $|e^z|$ is smaller than $1$ if $\text{Re}(z)$ is nonpositive
Question : Given $\text{Re}(z) \le 0$ prove that $|e^z| \le 1$.
Try:
$z=x+yi$, it's given that $x \le 0$.
$$|e^{z}| = |e^{x+yi}|=|e^xe^{yi}|=e^x|e^{yi}|,$$ with $e^x \le e^0$ because $f(x)=e^x $ is ...
1
vote
2answers
43 views
Polar coordinates - issue with direction denoted by angle
Convert $1-\sqrt{3}i$ to polar coordinates $(r,\varphi)$.
I started by computing $r=|1-\sqrt{3}i|=\sqrt{1^2+\sqrt{3}^2}=\sqrt{4}=2$. When I tried to compute the angle I did something like
...
0
votes
1answer
67 views
Geometric meaning of $Arg(\frac{z_{1}}{z_{2}})=2Arg(\frac{z_{3}-z_{1}}{z_{3}-z_{2}})+2\pi k$ for $z_{i}$ on the unit circle
I was given the following question:
Let $z_{1},z_{2},z_{3}\in\mathbb{C}$ s.t $|z_{1}|=|z_{2}|=|z_{3}|=1$,
it is known that
$Arg(\frac{z_{1}}{z_{2}})=2Arg(\frac{z_{3}-z_{1}}{z_{3}-z_{2}})+2\pi
...
1
vote
1answer
39 views
If $(a_n)_n$ is bounded, then $\sum\limits_n a_{n}n^{-z}$ converges uniformly for $\Re z \geq 1+\epsilon$
More specifically, let $(a_n)_n$ be a bounded sequence of complex numbers. Show that for each $\epsilon>0$, the series $\sum\limits_{n=0}^{\infty} a_{n}n^{-z}$ converges uniformly for $\Re z \geq ...
0
votes
2answers
231 views
Show that $\sum \frac{z^k}{k}$ does not converge uniformly for |z|<1
It's easy to see that the series $\sum \frac{z^k}{k}$ converges locally for |z|<1, by comparison with $\sum z^k$. I'm trying to show why it doesn't converge uniformly. Would it be correct to say ...
