2
votes
1answer
30 views

How to show $\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$

How to show $$\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$$ I begin with $$\sin(x+iy) = \frac{e^{x+iy}-e^{-x-iy}}{2i} = \frac{e^xe^{iy}-e^{-x}e^{-iy}}{2i}$$ $$ = ...
0
votes
1answer
25 views

Plot of a domain in the complex plane

I am trying to plot the following domain in the complex plane: $\lbrace x\in\mathbb{C}|\: |x^{2}-1|<r\rbrace$ for some $r>1$. I know that in general to take a square root of a complex number ...
0
votes
1answer
55 views

Complex number, strangely written

Find all the complex solutions of the equation: $$\frac{z^3}{i} = 1$$ I mean is this the same thing as $$z^3 = i$$? Because I don't understand why my teacher would put it like that on a test. At ...
0
votes
2answers
30 views

Finding numbers $a$ and $b$ for a complex number

Problem. Given a complex number $$z=2-2i$$ Find numbers $a$ and $b$ such that $$a+ib = \frac{1}{z}$$ I tried multiplying both sides by $z$ and got $$(a+ib)(2-2i)$$ $$= 2a-2ai+2bi-2bi^2$$ ...
2
votes
1answer
26 views

Effect of sigmas inequality on sequences

We have two nets of complex numbers $\{z_\alpha\}_{\alpha\in I},\{w_\alpha\}_{\alpha\in I}$ for some set $I$ which might be uncountable, and we have $$\sum_{\alpha\in I}|w_\alpha|\leq ...
1
vote
3answers
58 views

How to find out the real part of this.

I have to sum this: $$S:=\cos\left(\frac{\pi}{M+1}\right)+\dots+\cos\left(\frac{M\pi}{M+1}\right)$$ Where $M$ is a given natural number. I tried with this: Since $$e^{i a}=\cos a+i\sin a$$ and ...
0
votes
1answer
50 views

Find the residues of $f(z) = \left( \frac{z-1}{z+1}\right)^{\frac{1}{2}}\frac{1+z} {1+z^{2}}$

Consider the function $$f(z) = \left( \frac{z-1}{z+1}\right)^{\!\frac{1}{2}}\frac{1+z} {1+z^{2}}$$ I want to calculate the residues of $f$ in $\{+i,-i\}$. Using the usual techniques, we have that ...
4
votes
3answers
91 views

Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
1
vote
0answers
40 views

Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
2
votes
1answer
55 views

A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
9
votes
2answers
135 views

Maximum of $\frac{\sin z}{z}$ in the closed unit disc.

I have some trouble with the following question: Let $$f(z)=\frac{\sin z}{z},\quad\text{for }z\in\mathbb{C}.$$ What is the maximum of $f$ in the closed unit disc ...
1
vote
1answer
140 views

Proof that imaginary numbers exist? [duplicate]

How do imaginary numbers exist? I know you can't use the conventional number system, but use the complex one. But, how do you prove that the complex number system exists in the first place?
3
votes
3answers
132 views

How does $\mathrm {e}^z$ and $\log z$ look like as complex functions.

I want to visualize complex functions $\mathrm e^z$ and $\log z$ in $C$, here $z\in\Bbb C$. I want to know their behavior and zeros and singularities. Can anyone explain me in an easy way. Thank you ...
3
votes
1answer
228 views

Zeros of complex function sequence (Application of Rouche's Theorem).

For a given sequence of complex functions: $\phi_n(z)= 1+\frac1n-z-e^{-z}$; here $z\in${$z| Rez>0$}. I want to prove that : (1). $\phi_n $ has a unique zero $z_n$ in the half plane. (i.e. there ...
3
votes
1answer
69 views

How to evaluate the summation $S_b$

This question is from my notebook, not hw or else, only exercise to understand better. I tried by myself. However, since my trail are too trivial, I dont need to write here. i am confused a bit. I ...
1
vote
3answers
99 views

Complex book suggestions

I take complex analysis course. And my instructor use -Bak and Newman's complex analysis book, springer. This book explains too fast and superficially. Please give me book suggestions which are the ...
1
vote
1answer
41 views

Complex function with two inverses?

I was computing the inverse of the complex function $$\xi(z) = z + \frac{1}{z} \quad \text{ where } z \ne 0$$ which lead me to a strange conclusion. If we set $$\xi(\xi^{-1}(z)) = z$$ and solve ...
1
vote
0answers
51 views

Regarding Logarithmic of complex numbers

$\newcommand{\Log}{\operatorname{Log}}$ In my undergraduate complex analysis textbook, it claims that $$\Log(1+i)^2=2\Log(1+i)$$ I am not sure if this is a misprint or there is actually a way of ...
3
votes
1answer
117 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
1
vote
1answer
64 views

what are real and imaginary part of this expression

I have $M:=\sqrt{\frac{a\cdot(b+ic)}{de}}$ and all variables $a,b,c,d,e$ are real. Now I am looking for the real and imaginary part of this, but this square root makes it kind of hard.
4
votes
3answers
505 views

Type of singularity of $\log z$ at $z=0$

What type of singularity is $z=0$ for $\log z$ (any branch)? What is the Laurent series for $\log z$ centered at 0, if exist? If the Laurent series has the form $\sum_{k=-\infty}^{\infty} a_kx^k$, ...
1
vote
1answer
77 views
2
votes
0answers
82 views

First time dealing with limits with complex numbers in it.

I am solving the following problem. Investigate the behavior (convergence of divergence) of $\Sigma a_n$ if $$a_n = \frac{1}{1+z^n}, \quad \text{ for } z \in \Bbb C.$$ First of all, I am ...
0
votes
0answers
33 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}\ |\; ...
-1
votes
2answers
42 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 ...
36
votes
5answers
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
2answers
54 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
1answer
59 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 :)
2
votes
1answer
53 views

Is this estimation correct?

I have to estimate the following quantity $$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$ in $\mathbb{R}^3$ ($\lambda>0$) where ...
0
votes
0answers
52 views

Limit points of a sequence contained in $S^1$

Let $\theta\in (0,2\pi)$ be a real number such that $\displaystyle\frac{\theta}{\pi}\notin\mathbb{Q}$. We define $z:=\cos(\theta)+i\sin(\theta)\in S^1\subseteq\mathbb{C}$ and let ...
1
vote
1answer
95 views

Where is mapped a strip by complex square root?

Does the complex square root maps a strip such this $$\lbrace {z|a\leq \Re z\leq b, 0<\Im z<1\rbrace}$$ $a,b>0$, into another strip?
2
votes
1answer
58 views

Problem on property of complex polynomials

Given that, the polynomial $f(z) = z+a_2 z^2 + \ldots + a_n z^n$ is one-to-one from the open unit disc $D$ to $\mathbb{C}$. I have to show that $|na_n|\leq 1$. I have tried to show this by ...
1
vote
1answer
244 views

Complex Analysis and Limit point help

So S is a complex sequence (an from n=1 to infinity) has limit points which form a set E of limit points. How do I prove that every limit point of E are also members of the set E. I think epsilons ...
1
vote
1answer
76 views

Lower and upper bounds for fractional linear transformation

If we have $k(z)=\frac{z}{1-tz}$ which is convex in unit disk, then $k(\bar{z})=\overline{k(z)}$, $k(z)$ maps real axis to real axis where $|z|\leq{r}$, $t\in\mathbb{R}$. What is the upper and lower ...
4
votes
2answers
108 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
2
votes
0answers
119 views

Do I understand complex differentiability correctly?

I don't use complex analysis much in my "day job", but I'm perusing the materials for personal interest. I just wanted to check with the community about my understanding of complex analytic ...