Tagged Questions
5
votes
1answer
57 views
Why is the bailout value of the Mandelbrot set 2?
For the past few days I've been studying the Mandelbrot set, and many say that if the iterations of a point stay within a magnitude of 2, the point converges. A very natural question of "why is the ...
2
votes
1answer
27 views
Inequality holding for complex numbers in the unit disk
In Nehari's book Conformal Mapping he gives it as an exercise to prove that for $a,b\in \mathbb{C}$, $|a|, |b| <1$ we have $$\frac{|a|-|b|}{1-|ab|} \leq \left|\frac{a-b}{1-\overline{a}b}\right| ...
6
votes
4answers
147 views
If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?
Is the following true if $\theta\in\mathbb{Q}$?
$$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$
Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
1
vote
1answer
44 views
If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Is $f(x+iy) = \exp(x+iy)$?
$(1)$ If $f$ is an entire function such that $f(iy) = \exp(iy)$ where $0 \leq y \leq 1$. Then, is $f(x+iy) = \exp(x+iy)$ for every $x$ and every $y$?
$(2)$ If $f$ is an entire function such that ...
3
votes
2answers
62 views
Finding two eigenvalues which add to $1$
$\textbf{Question}$: For $0<t<\pi$, the matrix
$$
\left( \begin{array}{cc}
\cos t & -\sin t \\
\sin t & \cos t \\
\end{array}
\right)
$$
has distinct complex eigenvalues $\lambda_1$ ...
2
votes
1answer
61 views
Entire function $f$ such that $|f(z)| ≤ K |z|^3$ for $|z|\ge1$ and $f(z) = f(iz)$ for every $z∈C$
Let $f(z)$ be an entire function such that for some constant $K$ , $|f(z)| ≤ K$ $|z|^3<$ for $|z|\ge1$ and $f(z) = f(iz)$ $∀z∈C$ , then which of the following are correct ?
(A)$ f(z) = Kz^3$ $∀ ...
1
vote
1answer
29 views
Describe the graph locus represented by this equation
I want to know the shape of the region described by
$$ Im(z^2) = 4 $$
so I did the following:
$$ z=x + iy $$
$$z^2 = x^2 + 2xiy -y^2 $$then
$$Im(z^2) = 2xy
$$
then the locus is $$ 2xy = 4 $$
...
0
votes
2answers
51 views
Complex number question
For any complex numbers $z_1, z_2$, is the quantity $S$: $$
S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 ...
2
votes
1answer
56 views
Is there a geometric relationship between plane geometry and polynomials?
It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
0
votes
2answers
36 views
contradicting identity theorem?
the identity theorem for holomorphic functions states: given functions $f$ and $g$ holomorphic on a connected open set $D$, if $f = g$ on some open subset of $D$, then $f = g$ on $D$
Let $f(z) = \sin ...
5
votes
2answers
95 views
Prove that the zeros of an analytic function are finite and isolated
Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$.
The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
3
votes
3answers
121 views
Solve $\sin(z) = z$ in complex numbers
Show that $\sin(z) = z$ has infinitely many solutions in complex numbers.
Little Picard theorem should help, but using big Picard theorem is undesirable.
Thanks a lot!
0
votes
1answer
143 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 ...
5
votes
1answer
102 views
A case where $z^z = 0$ where $z$ is complex number
Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
-2
votes
2answers
71 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 ...
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
$$\large \rm ...
1
vote
2answers
54 views
determining residue for the purposes of calculating an integral
Determine the integral
$$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$
using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications.
In order to do this. We ...
0
votes
1answer
20 views
convergence of complex series
Set that Re $z_n>=0$,$\forall$ n $\in$ N,Proof that if $\sum z_n$ and $\sum {z_n}^2$ are both convergent,then $\sum |z_n|^2$ is also convergent.
Well I've no idea how to tackle it.
3
votes
4answers
121 views
When does $az + b\bar{z} + c = 0$ represent a line?
$a,b,c$ and $z$ are all complex numbers. My idea was to show that it passes through the point $\infty$ in the extended complex plane, but I'm not quite sure how to execute that.
Update:
It says in ...
0
votes
3answers
49 views
Problem on Complex Numbers
Which of the following is most correct for the complex numbers Z and W, marked with "x" in the picture of the complex numbers below? (the dashed circle represents the unit circle)
a) $Z = W + 3i$
b) ...
0
votes
0answers
52 views
Does it make sense to talk about $ O(z)$ if $z$ complex?
Does it make sense to talk about $ O(z)$ if $z$ complex? I would have thought that the usual definition wouldn't hold, since doesn't the fact that we don't have an order on $\mathbb{C}$ change things? ...
9
votes
6answers
576 views
inequality involving complex exponential
Is it true that
$$|e^{ix}-e^{iy}|\leq |x-y|$$ for $x,y\in\mathbb{R}$? I can't figure it out. I tried looking at the series for exponential but it did not help.
Could someone offer a hint?
0
votes
1answer
58 views
Taylor Series Expansion with e and sin
Show that when $z\neq0$,
(a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$
(b) ...
2
votes
2answers
50 views
Proving a complex equality
Let $a$ and $c$ be complex numbers. Show there exists complex numbers $z$ s.t. $|z-a|+|z+a| = 2|c|$ if and only if $|a| \leq |c|$.
I've shown the forward direction correctly, but I don't know how to ...
0
votes
1answer
69 views
Form of periodic function involving exponential
I am trying to prove that if the function $f(z)= a_{1}e^{\lambda_{1}z} + ... + a_{n}e^{\lambda_{n}z}$ is periodic of period $T \neq 0$ with $a_{i} \neq 0$ for every $i$, then $\lambda_{i} = 2k_{i}\pi ...
1
vote
3answers
40 views
Sum of powers of $z$ with $|z| = 1$, $z \neq 1$
I am trying to prove that for $z \neq 1$ with $|z| = 1$ the sequence of partial sums $s_{n}= \sum_{k=0}^{n} z^{k}$ is bounded. I kinda understand the reason why it should happen but I do not see the ...
0
votes
0answers
25 views
Fractional linear transformations with given properties
I need a function of the form
$\displaystyle f(z):= \frac{az+b}{cz+d}, \qquad z\in\mathbb{C}-\{-\frac{d}{c}\}, \qquad ad-bc\neq0$
which carries the half-plane $\{z\in\mathbb{C}\ |\; ...
2
votes
3answers
66 views
Incoherence using Euler's formula
Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = ...
3
votes
1answer
109 views
Can't prove this limit of complex numbers from a paper
Okay so I found in a paper, marked as "simple exercise", the following thing: for $z,b \in \mathbb{C}$,
$$\lim_{b\to0} \frac{1}{z-b} + \frac{1}{2b} - ...
0
votes
3answers
59 views
Simplifying $i\cdot(\arctan(\frac{y}{x})+\arctan(\frac{x}{y}))$ where $z=x+iy$
In one of my calculations I arrived at the expression $$i\cdot(\arctan(\frac{y}{x})+\arctan(\frac{x}{y}))$$
I know that $$\arctan(\frac{y}{x})=Arg(z)$$ is there something similar
for ...
1
vote
1answer
48 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
1answer
43 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
$$\vert e^{i\Re z\vert ...
1
vote
1answer
63 views
Continuous function on simple closed contour
Let $f$ denote a function that is continuous on a simple closed contour $C$. Using the Cauchy Integral formula, prove that the function $g(z)=\frac{1}{2\pi i}$ $\int_C$ $\frac{f(s)ds}{s-z}$ is ...
1
vote
1answer
35 views
Connected set on complex plane
What's the numebr of connected components for the set of complex numbers $\{e^z:|z|=1\}$ on the complex plane?
Remark: It represents a simple closed curve which intersects the real axis at points ...
3
votes
0answers
60 views
If the product of two analytic functions is zero, then one must be identically zero.
I want to prove this statement:
Let $f,g$ be analytic on $D(0,2)$. If $f(z)g(z) = 0$ when $z = 1/n$ for $n \in \mathbb{N}$, then either $f \equiv 0$ or $g \equiv 0$ in $D(0,2)$.
My attempt:
...
0
votes
3answers
44 views
Complex Sine bounded when $|\text{Im } z| < 1$
Prove that $\sin z$ is bounded on $\{z \in \mathbb{C} : |\text{Im } z| < 1\}$.
I know how to prove that it is unbounded, but I'm stuck on this.
1
vote
3answers
61 views
Adding real and imaginary parts
When trying to add $x$ to $x^{*}$ is it allowed to say that it would be equal to $2|x|$ i.e. so that $$x+x^{*}=2|x| $$ If this isn't the case is there any way to add them or should they be left as ...
4
votes
1answer
192 views
Rudin Theorem 1.35 - Cauchy Schwarz Inequality
Any motivation for the sum that Rudin considers in his proof of the Cauchy-Schwarz Inequality?
Theorem 1.35 If $a_1,...,a_n$ and $b_1, ..., b_n$ are complex numbers, then
...
5
votes
1answer
52 views
Roots of a polynomial and its derivative
All roots of a complex polynomial have positive imaginary part. Prove that all roots of its derivative also have positive imaginary part.
It's not a homework. This issue has been proposed in the ...
-1
votes
2answers
82 views
How to integrate complex exponential??
Consider
$$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$
Why do we only look at the real part? What about the imaginary part ...
0
votes
1answer
67 views
Imaginary complex numbers
Let $z=x+iy$ and $v=2xy$, show that $v=Im[z^2]$ and find a harmonic conjugate of $v$ on domain $D$. Also find the largest domain $D$ on which $v$ is harmonic.
33
votes
6answers
1k views
Why do we negate the imaginary part when conjugating?
For $z=x+iy \in \mathbb C$ we all know the definition for the "conjugate" of $z$, $\bar{z}=x-iy$. Geometrically this is the reflection of $z$ across the $y$ axis.
My question is: couldn't we have ...
2
votes
5answers
86 views
De Moivre's Theorem Related - Complex number
According to de Moivre's Theorem: If $n$ is any positive integer then: $(\cos\theta + i\sin\theta)^n=\cos n\theta +i\sin n\theta$
Also $(\cos\theta +i\sin \theta)^{\frac{1}{n}} = \cos \frac{2r\pi ...
0
votes
1answer
85 views
How can I show that circles in the complex plane correspond to circles on the Riemann sphere? How about lines?
Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is
a. a circle if $ T $ is a circle.
b. a circle minus (0, 0, 1) if $ T $ is a line.
Here we are defining $ ...
2
votes
2answers
41 views
Holomorphic function $f$ such that $f'(z_0) \neq 0$
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an holomorphic function such that $f'(z_0) \neq 0$ for some $z_0 \in \mathbb{C}$.Prove that there is $r>0$ such that, if $|z-z_0|<r$ and $z \neq z_0 ...
1
vote
2answers
48 views
complex number power
I have question related to power of i,which is determined by equality $i=\sqrt{-1}$
actually from complex number book I know that $i^2=-1$, as much as i know if we compare physical ...
1
vote
3answers
87 views
Find the principal argument of a complex number
I have a text book question to find the principal argument of
$$ z = {i \over -2-2i}. $$
I know formulas where we find using $$ \tan^{-1} {y \over x}$$
but I am kinda stuck here can somebody please ...
3
votes
3answers
76 views
Proving that a complex number $z$ is real.
A problem I have in my book is to prove that $z$ is real if and only if $\bar{z} = z$.
So far I have got that for $z = x + iy$, if $z$ is real, $y = 0$ and thus $z = x = \bar{z}$ as $\bar{z} = x - ...
2
votes
2answers
51 views
Complex numbers
I am a newbie to complex numbers so please bear with me if i ask some very naive question.,
So i was trying to solve my class tutorials and the very first question is,
Show that ...
