Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
4
votes
2answers
90 views
Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $
I'm trying to find
$$\sum _{k=1}^{n} k\cos(k\theta)\qquad\text{and}\qquad\sum _{k=1}^{n} k\sin(k\theta)$$
I tried working with complex numbers, defining $z=\cos(\theta)+ i \sin(\theta)$ and using ...
1
vote
4answers
106 views
Fourier Analysis
I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function?
Can someone provide me simple information ...
3
votes
1answer
53 views
$c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$
Let $c$ is complex-number satisfying :
$(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$
So, how could i get
$(3c^{100}+\frac{2}{c^{100}}+1)(c^{100}+\frac{2}{c^{100}}+3)$ ?
1
vote
1answer
36 views
Lower bound for polynomial with complex coefficient
Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$.
...
0
votes
1answer
25 views
Residue theorem for a rational integral
given the integral
$$ \int_{0}^{\infty}dx \frac{x^{s}}{Q(x)} = \oint _{C} dz \frac{z^{s}}{Q(z)} $$
here 's' is a parameter , this can be real or complex so there is a branch cut
'C encloses ALL the ...
1
vote
1answer
42 views
Complex proof (with sum)
How can I prove $$\sum_{n=-\infty}^∞\frac{1}{(a+bn)^2}=\frac{π^2}{b^2} \csc^2 \frac{πa}{b}$$
When $\frac{a}{b}\notin$*Z*?
0
votes
2answers
36 views
Complex expression for periodic binary sequences
We have infinite binary sequences of type
$$\langle g_n \rangle_{j=4}=\{0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,...\} \,;\, n\to\infty$$
where $j$ indicates the length of a period that starts/ends with a $1$; ...
1
vote
1answer
42 views
Rotation on the unit circle K
If $\{a^{n}:n\in\mathbb{Z}\}$ is dense on the unite circle K, then $\{a^{n}:n< 0\}$ is also dense on K. How to prove this result?
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?
5
votes
5answers
170 views
What is the square root of $i^4$?
What is the $\sqrt{i^4}$?
$i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$?
Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$?
When I plug it into my TI-89 Titanium, I get $1$.
Edit: I ...
1
vote
1answer
57 views
Find a complex number that satisfies the equation
Find one complex value
of $x$ that satisfies the equation
$\sqrt{3}\cdot x^7+x^4 +2=0$
0
votes
1answer
33 views
Complex plane sets - domain |Argz | < pi/4
Why is the following complex set a domain:
|Arg z | < pi/ 4 if Arg z = 0 is not defined, so there is no polygonal path between the two quadrants.
-1
votes
2answers
39 views
Estimate a complex modulus
I have to estimate the following quantity
$$\vert e^{iz\vert x\vert}-e^{i\lambda\vert x\vert}\vert^2$$
where $x\in\mathbb{R}^3$, $\Im z>0$ and $\lambda>0$.
So I write
$$\vert e^{i\Re z\vert ...
0
votes
1answer
29 views
Put the following in rectangular form.
$$(\sqrt{3}+i)^7$$
My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate.
Thank you.
1
vote
1answer
47 views
Continuous function on simple closed contour
Let $f$ denote a function that is continuous on a simple closed contour $C$. Using the Cauchy Integral formula, prove that the function $g(z)=\frac{1}{2\pi i}$ $\int_C$ $\frac{f(s)ds}{s-z}$ is ...
0
votes
1answer
40 views
Contour Integrals and positively oriented circles
If $C_0$ denotes a positively oriented circle $|z-z_0|=R$, then $\int_{C_0}$ $(z-z_0)^{n-1} dz$ = $\left\{
\begin{array}{lr}
0 & n=\pm1, \pm2, ...\\
2\pi i & n=0\\
...
1
vote
1answer
32 views
Connected set on complex plane
What's the numebr of connected components for the set of complex numbers $\{e^z:|z|=1\}$ on the complex plane?
Remark: It represents a simple closed curve which intersects the real axis at points ...
0
votes
2answers
42 views
Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where $x,y$ complex numbers?
Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where x,y are complex numbers? Or is it simply solved by observation, and so the answer is $x=-i$ and $y=1$?
Thanks in advance.
3
votes
0answers
51 views
If the product of two analytic functions is zero, then one must be identically zero.
I want to prove this statement:
Let $f,g$ be analytic on $D(0,2)$. If $f(z)g(z) = 0$ when $z = 1/n$ for $n \in \mathbb{N}$, then either $f \equiv 0$ or $g \equiv 0$ in $D(0,2)$.
My attempt:
...
0
votes
3answers
39 views
Complex Sine bounded when $|\text{Im } z| < 1$
Prove that $\sin z$ is bounded on $\{z \in \mathbb{C} : |\text{Im } z| < 1\}$.
I know how to prove that it is unbounded, but I'm stuck on this.
0
votes
1answer
69 views
Why is the second equality wrong?
Here's a "proof" of $e^x=1$ for all $x$: $$\exp(x)=\exp\left(i2π⋅\frac{x}{i2π}\right)=\bigl(\exp(i2π)\bigr)^{x/(i2π)}=1^{x/(i2π)}=1$$ Why is the second equality wrong?
1
vote
2answers
59 views
Complex numbers - finding minimum value
For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$ , find the minimum value of $|z_1-z_2|$
Can we go like this :
Let $z_1 = x +iy$ therefore $|z_1| = \sqrt{x_1^2+y_1^2}$ and ...
3
votes
1answer
48 views
Compact sets of the complex plane with countable boundary
Suppose $E$ is a compact set of the complex plane and the boundary of $E$ is a countable set. How does one prove that $E$ is equal to its boundary?
2
votes
3answers
54 views
Distance between point and line in the complex plane
Let $a,b$ be fixed complex numbers and let $L$ be the line
$$L=\{a+bt:t\in\Bbb R\}.$$
Let $w\in\Bbb C\setminus L$. Let's calculate $$d(w,L)=\inf\{|w-z|:z\in L\}=\inf_{t\in\Bbb R}|w-(a+bt)|.$$
The ...
0
votes
1answer
48 views
Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.
I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
1
vote
2answers
71 views
is $1^z=1$ for all complex values of $z$?
i would like to see if $1^z=1$ is valid for all complex variable $z$,first of all you can rewrite above equation as
$1^{a+b*i}=e^0$
here i think that instead of $+$ sign, we may take take ...
0
votes
0answers
21 views
check validity of following manipulation
in my algebra book,there is written following well known identity
$e^{2*\pi*i}=1$
generally we can use also this identity $e^{k*\pi*i}=(-1)^k$
and if instead of $k$,we put $2$ we get ...
1
vote
1answer
42 views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k ...
1
vote
0answers
30 views
Compare one real number to one complex number. [duplicate]
I understand that complex numbers can be neither ordered nor compared by 'size', but if mapped one for one by a transformation, then they can be. Latter point aside, can I say that $2-xi < 2 < ...
1
vote
5answers
74 views
If $z$ is a complex number of unit modulus and argument theta
If $z$ is a complex number such that $|z|=1$ and $\text{arg} z=\theta$, then what is $$\text{arg}\frac{1 + z}{1+ \overline{z}}?$$
1
vote
3answers
203 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
32 views
How can I calculate the bode magnitude and frequency as well as their plots?
I've been trying to figure this problem out for a while now. I've been given a transfer function $$H(s) = \frac{s(s+100)}{(s+2)(s+20)}.$$ I'm supposed to calculate the bode magnitude and frequency for ...
1
vote
3answers
60 views
Adding real and imaginary parts
When trying to add $x$ to $x^{*}$ is it allowed to say that it would be equal to $2|x|$ i.e. so that $$x+x^{*}=2|x| $$ If this isn't the case is there any way to add them or should they be left as ...
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
$$
\begin{align}
x+y+z & =1\\
x^2+y^2+z^2 & =2\\
x^3+y^3+z^3 & =3
...
0
votes
0answers
45 views
Show the steps from step 1 to step 2 (2D Potential - Physics + Maths)(Really Urgent)
I have been trying to show
$$ 2 \operatorname{Im}\left( \sum_{n=1}^{\infty} \dfrac{z^n}{n} \right) = \tan^{-1} \left( \dfrac{\sin(\pi x / a)}{\sinh(\pi y / a)} \right) $$
where $$ \large{z = e^{i ...
0
votes
1answer
28 views
How to find the phase of a complex-valued function not in trigonometric form?
I have the following function:
$$(1 + (jw/w_i)^2 - 2j\alpha_i(w/w_i))\over(1 + (jw/w_i)^2 + 2j\alpha_i(w/w_i))$$
I know that the magnitude is 1 since this is a ratio of complex conjugates, but how ...
0
votes
1answer
37 views
Determinant formula and invertibility.
I am working on a problem where I need to find the determinant of
$$
\begin{bmatrix}
b & a & & \\
& b & a \\
& & & \ddots \\
& & & & ...
4
votes
1answer
152 views
Rudin's Proof of Cauchy-Schwarz (Theorem 1.35)
Any motivation for the sum that Rudin considers in his proof of the Cauchy-Schwarz Inequality?
1.35 Theorem If $a_1,...,a_n$ and $b_1, ..., b_n$ are complex numbers, then
$|\sum_{j=1}^n ...
2
votes
2answers
75 views
Finding the number of different ordered quadruples $(a,b,c,d)$ of complex numbers
Find the number of different ordered quadruples $(a,b,c,d)$ of complex numbers such that:
$$a^2=1$$
$$b^3=1$$
$$c^4=1$$
$$d^6=1$$
$$a+b+c+d=0$$
3
votes
3answers
119 views
A general proof of $f\left(\bar{z}\right)=\overline{f\left(z\right)}$
As a school student I have seen a striking property of functions .
$$f\left(\bar{z}\right)=\overline{f\left(z\right)}$$
Where $z$ is a complex number and $\bar{z}$ it's complex conjugate. For eg: ...
1
vote
2answers
41 views
Sin inverse of a complex number
Is it possible to calculate the value of $\delta$ from the relation
$\delta=\sin^{-1}(5.4i)$ ? where $i=\sqrt{-1}$
1
vote
1answer
50 views
$\int_{-\infty}^{\infty}i\cdot \sin(x)\sin(2{\pi}kx)\;dx$ during Fourier transform
I am trying to do a time-to-frequency domain transform using Fourier transform. My function is very simple:
$$
f(x) = \sin(x)
$$
By definition its Fourier transform should be:
$$
F(k) = ...
1
vote
2answers
35 views
Help please on complex polynomials
I wanted to know if there's any good approaches to these questions
a)By considering $z^9-1$ as a difference of two cubes, write $1+z+z^2+z^3+z^4+z^5+z^6+z^7+z^8$
as a product of two real factors one ...
1
vote
2answers
42 views
Finding roots of complex equation?
Determine all roots of the equation $x^6+(3+i)x^3 + 3i = 0$ in $\mathbb{C}$ Express answers in standard form
0
votes
2answers
54 views
Prove the direct product of nonzero complex numbers under multiplication.
Let $\mathbb{C}^{\times}$ be the group of nonzero complex numbers under multiplication. Then $\mathbb{C}^{\times}$ is the direct product of the circle group $T$ of unit complex numbers and the group ...
3
votes
3answers
52 views
Solve $z^5 + 16\bar z = 0$ for $z\in \mathbb{C}$
Solve $z^5 + 16\bar z = 0$ for $z\in \mathbb{C}$.
Need some help figuring out this problem.
3
votes
1answer
43 views
Proof required that $\sum _{n=1} ^N(1-e^{(2n+1) \pi i/N})^{-1} = \frac N 2$
Numerical evidence suggests this is true, for all natural numbers $N$:
$\sum _{n=1} ^N(1-e^{(2n+1) \pi i/N})^{-1} = \frac N 2$
Can anyone prove it?
5
votes
1answer
47 views
Roots of a polynomial and its derivative
All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part.
It's not a homework. This issue has been proposed in the ...
0
votes
1answer
51 views
Two quick eigenvalues & complex numbers questions
A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ?
($Im(v)$denoting the imaginary part of the vector $v$)
My understanding: since every row of the vector is a complex number (say ...
7
votes
5answers
158 views
If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle
Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.)
How do I prove this? I tried to use the dot product of 2 adjacent ...
