Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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0
votes
1answer
15 views

Complex numbers: find all $z$ such that $e^{z-2}=-ie^2$

Ok, so I think I'm getting the hang of this. Is this more or less on the right track? $$e^{z-2}=-ie^2$$ $$e^ze^{-2}=-ie^2$$ $$e^z=-ie^4$$ $$\ln(e^z)=\ln(-ie^4)$$ $$z=\ln|-i|+iarg(-i)+2\pi ik+4$$ ...
0
votes
0answers
13 views

If $c_{n}$ coefficient of the expansion of $f$. Show that $\sum_{n=0}^{\infty}\left|c_{n}\right|^{2}r^{2n}$ is an expression determined by an integral

Let $U\subseteq \mathbb{C}$ be an open set, $z_{0}\in U$ and $R>0$ such that $\mathcal{B}_{R}(z_{0}) \subseteq U$. Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function with Taylor's serie ...
0
votes
1answer
22 views

Basic Complex Analysis Question: simplification of $e^{2\pi i\left(17-\frac{1}{3}\right)}$

I have the following detail of a derivation: $$ \begin{align} [\dots] &= 2^{50}\cdot e^{2\pi i\left(17-\frac{1}{3}\right)} \\ &= 2^{50}\cdot e^{-\frac{2\pi}{3} i} \end{align} $$ See source ...
5
votes
4answers
42 views

Find all $z$ such that $e^z=6i$

Am I on the right track to solving this? $$e^z=6i$$ Let $w=e^z$ Thus, $$w=6i$$ $$e^w=e^{6i}$$ $$e^w=\cos(6)+i\sin(6)$$ $$\ln(e^w)=\ln(\cos(6)+i\sin(6))$$ $$w=\ln(\cos(6)+i\sin(6))$$ ...
2
votes
1answer
30 views

What is the difference in notation for Ln(z) vs. ln(z)

We are learning about the principal vs. non-principal values of the natural logarithm. There are 2 problems: $\ln(-7)$ $\operatorname{Ln}(-7)$ I solved $\ln(-7)$ for: $$=\ln|-7|+i\pi+2\pi k$$ Is ...
0
votes
1answer
12 views

Example of maximum modulus principle

As it's known , an holomorphic($\neq constant$) function $f:G\subseteq\mathbb{C}\rightarrow \mathbb{C}$ has maximum modulus on $\partial G$ . I wuold an example of a function holomorphic on a disk ...
-2
votes
0answers
19 views

Complex numbers - proving properties of conjugates [on hold]

How do you prove this property: For all $z$ is an element of $C$: $z + \bar{z}$ is an element of $\mathbb{R}$?
1
vote
4answers
32 views

Complex numbers - solving equation in $\mathbb{C}$

How can you solve the following equation in the set of complex numbers? $$\frac{z- 1}{i-1} + \frac{z-2}{i-2} + \frac{z-3}{i-3}= 0$$
3
votes
4answers
47 views

What can be said about the relationship between the complex numbers $\lvert z\rvert^n$ and $\lvert z^n\rvert$?

I've been playing around with this for a while without much progress. More precisely, I suppose, I'd like to know if one always less than or equal to the other? The fact that one never sees this in ...
2
votes
1answer
28 views

Geometrically interpreting complex numbers.

Prove that $|e^{i \alpha} - e^{i \beta}| |e^{i \gamma} - e^{i \delta}| + |e^{i \beta} - e^{i \gamma}| |e^{i \alpha} - e^{i \delta}| = |e^{i \alpha} - e^{i \gamma}| |e^{i \beta} - e^{i \delta}|$ ...
4
votes
2answers
93 views

What is the derivative of $z^{-1}$ with respect to $\bar{z}$?

I asked a question here a few days ago but it wasn't answered and, as often happens with me, in trying to answer it myself I just confused myself out of understanding what I thought I knew. What is ...
1
vote
1answer
28 views

Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
2
votes
1answer
35 views

Imaginary number and absolute value integral - Fourier transform

I came across this integral problem: $$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$ Now I know how to integrate simple absolute value functions like: $\int_{-2}^{4}|x-2| dx$, we just ...
0
votes
2answers
40 views

Prove the inequality regarding complex numbers

If $\theta_i\in [0,\pi/6],i=1,2,3,4,5$.And $$\sin \theta_1\ z^4 + \sin\theta_2 \ z^3 + \sin\theta_3 \ z^2 + \sin\theta_4 \ z + \sin\theta_5=2$$ Prove that $|z|\gt \frac{3}{4}$.
0
votes
3answers
42 views

Complex Addition

Solving a question, I need to find the value of following in between the solution. $$(\frac{i+\sqrt3}{2})^{200} + (\frac{i-\sqrt3}{2})^{200}$$ The only useful thing I got was ...
2
votes
1answer
21 views

Exponential of a complex variable

Can someone please tell me if I am approaching this correctly? Given the following and asked to solve for the complex variable z: $$[e^z]^3-5e^z=0$$ My approach was purely algebraic and is why I have ...
0
votes
0answers
25 views

Which of the following sets is compact, bounded, closed or open and why? [on hold]

Which of the following sets is compact, bounded, closed or open and why? $M1= [-1,42]$ $M2= (-1,42]$ $M3= (-1,42)$ $M4= (-\infty, +\infty)$ $M5= \{z \in \mathbb C: 0 < \operatorname{Re} z + ...
1
vote
0answers
41 views

Conformal Mapping: Is this correct?

I have the following two circles in the complex plane, $z=x+iy$, which bound a region, $R$. The equations for the circles are given as follows: $$x^2+(y−1)^2=1\\x^2+(y−2)^2=4 $$ What I now want to do ...
2
votes
3answers
47 views

Does the complex modulus satisfy the power identity $|z^r|= |z|^r$?

Can we "split the modulus" of complex numbers? Let $z\in\mathbb{C}$. Then, does $$|z^{r}|=|z|^r$$ hold, where $r\in\mathbb{R}$. Is this true even for $r\in\mathbb C$ ? Also, can we show this? I am ...
0
votes
0answers
28 views

Do quaternions linearise tetration [on hold]

This is just a wild guess as I am not very familiar with quaternions or tetration. Sorry if my terminology is not quite right, I hope you get the general idea. Complex exponentiation is linear in ...
1
vote
1answer
38 views

Show that $\log(z)$ is real if z is real and positive.

Question The problem is this: Show that $\log(z)$ is purely imaginary (i.e. $\operatorname{Re\, Log} z$ $=$ $0$) if $|z|=1$. Show that $\log(z)$ is real if $z$ is real and positive ...
2
votes
2answers
44 views

Convert $e^z$ to Cartesian form (complex numbers)

Convert $e^z$ to $a+bi$ I'm having trouble figuring out this very simple problem. Below is my attempt, but can you really have $1/e$ as the modulus of a complex number? $$z=-1+\frac{i\pi}{4}$$ ...
7
votes
2answers
117 views

Solving $e^{\sin(z)}=1$ in the complex plane

I am trying to solve $e^{\sin(z)}=1$ in the complex plane. I know that this means that $\sin(z)=2k\pi i$ for some integer $k$. This is equivalent to saying that $$\frac{e^{iz}- e^{-iz}}{2i}=2k \pi ...
1
vote
1answer
15 views

Trouble with an inequality between magnitudes of complex numbers

We are supposed to show that $$|ab^* + a^*b| \leq 2|ab|$$ where a and ba re complex numbers and a* and b* are their respective conjugates (so $a = x_1+iy_1$, $a^* = x_1-iy_1$, $b = x_2+iy_2$, $b^* = ...
0
votes
0answers
44 views

Mapping circles in the complex plane

I have the following two circles in the complex plane, $z=x+iy$, which bound a region, $R$. The equations for the circles are given as follows: $$x^2+(y−1)^2=1 \\ x^2+(y−2)^2=4 $$ That is, I believe, ...
0
votes
0answers
20 views

Complex Space Factorization

I'm curious as to whether a visualization of the complex space and the area covered through various factorization methods has been put forth before. Each triangle in the mapping below represents a ...
0
votes
3answers
32 views

Prove that the four roots of unity form an abelian multiplicative group

My question is: Let $i = \sqrt{-1}$. Prove that the four roots of unity $\{1, -1, i, -i\}$ form an abelian multiplicative group. I know that abelian group is a group with commutative property. ...
-2
votes
1answer
34 views

Complex numbers- Proving when no variables are given [on hold]

How can I prove this $$\sin(\theta)-\cos(\theta) = \mathrm{cis}(\pi/2 - \theta)$$ I am sorry I could not show any attempts, I am absolutely stumped on how to do this.
1
vote
2answers
26 views

Product of terms involving complex exponents [on hold]

I have worked out the $\prod_{k = 1}^{50}i^k$. I get answer is $-i$. Is it correct?
2
votes
4answers
20 views

Summation of series involving complex exponents.

I am interested to find out what is $\sum_{k=0}^{200} i^k$, where $i$ is complex number
0
votes
1answer
64 views

Is $5 = -5$ according to this formula? [duplicate]

Today in maths class something occurred to me. How can this be: $\sqrt{(-5)^2} = ((-5)^2)^{1/2} = (-5)^{2(1/2)} = (-5)^1 = -5$ $\sqrt{(-5)^2} = ((-5)^2)^{1/2} = 25^{1/2} = 5$ This could ...
0
votes
0answers
23 views

Help with understanding when Log(z^k)=k Log(z) as well as drawing the function.

For the question I'm dealing with the property Log(z^k)=k*Log(z)in which I have to find the largest open set that this property is true when $k$ is a positive integer. I understand that this ...
3
votes
5answers
93 views

Give examples showing why $0\cdot \infty$, $\infty/\infty$, and $0/0$ are meaningless

Assuming arithmetic operations on $\overline{\mathbb{C}}$ (that's the extended complex plane) are defined via arithmetic operations on the corresponding sequences, I need to give examples showing why ...
1
vote
1answer
22 views

Image of Upper Half Disc under $w = 1/z$

I need to find the image of the upper half disc $|z|<1$, $Im\, z >0$ under the inverse transformation $w = 1/z$. Now, since $|z|<1$, $|z|^{2}<1$. Rewriting this as $z\overline{z}<1$, ...
1
vote
4answers
53 views

finding real roots by way of complex

I was given $$x^4 + 1$$ and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached. My teacher first found 4 ...
-1
votes
0answers
48 views

How to calculate with complex arguments?

I've got problems with calculating the complex argument of a number. We defined it like this: Let $$f: \Bbb C \ -{0} \to \Bbb [0;2π)$$ for $x=0$ we define no argument. For $zw≠0$ it is: $$Arg(zw)= ...
1
vote
1answer
29 views

Having trouble understanding(The proof of); $A \subset \mathbb{C}$ is finite implies the limit point set of $A$ is empty.

So I am self-studying Complex Analysis. The idea behind the proof is clear, and I understand the intuition etc. There are a few statements that are really giving me trouble accepting this proof ...
-1
votes
1answer
24 views

Is this sum differentiable?

Let $R$ be an infinite set of complex numbers $\rho$ with $0<\Re(\rho)<1$, $x$ be a nonzero real number and consider the summation $\sum_{\rho} x^{\rho}$. Is this sum differentiable with respect ...
-3
votes
1answer
45 views

Proving a fact about complex numbers [duplicate]

How to prove the following: For any complex numbers $z_1,...,z_n$, there exists a subset $E$ of $\{1,...,n\}$ s.t. $$\left| {\sum\limits_{j \in E} {{z_j}} } \right| \geqslant ...
3
votes
1answer
27 views

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$?

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$? Let $z=r(\cos(\theta)+i \sin(\theta))$. Then $\frac{\Re(z)}{|z|} = \frac{r \cos(\theta)}{r} = \cos(\theta) $; as the ...
0
votes
2answers
14 views

Manipulating exponents in complex numbers

I have had this problem for a long time now: Suppose we take the cube root of unity $\omega$. $\omega ^ 2 = e ^ {i {4\pi\over3}} = {(e ^ {i 4\pi})}^{1\over3} = {(e ^ {i 2\pi})}^{1\over3} = e ^ {i ...
0
votes
1answer
34 views

Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly continuous on $(-1,1)$

Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly continuous on $(-1,1)$. Partial proof : Suppose $f$ is uniformly continuous. $\implies \forall \epsilon > 0, \exists \delta ...
0
votes
0answers
23 views

Polar Equations (Complex)

So I'm trying to figure out what the angle $θ$ would equal at $x=-2$ for the polar equation $r=θ+sin(2θ)$. All I know is that $θ$ has a domain of $0\leθ\le \pi$ and $y < 1$ (pretty sure). I ...
2
votes
1answer
28 views

Why is this not a complex variable?

I'm watching a video about the Jacobi Theta Function on YouTube. At around 6:14 in the video, he has shown that... $$\vartheta (x) = \sum_{n\in \mathbb Z} e^{-\pi {n^2} x} = \sum_{k\in \mathbb Z} ...
-1
votes
1answer
55 views

How to solve this ${5z+5i}\over 2z^2+2$ complex number [closed]

I've tried solving this but i keep getting my answer wrong? I made the bottom all real numbers and multiplied it with the top as well. why is my answer wrong
0
votes
1answer
17 views

Show that the sets of isolated point of $E$ is a countable set - Axiome of choice

Let $E \subset \mathbb{C}$. Show that the sets of isolated point of $E$ is a countable set. That question is related to this question. However, my question somewhat different. Define ...
1
vote
3answers
38 views

If $|z-i|<1$ what can we deduce about $|z-1|$ and $|z+i|$

If $|z-i|<1$ what can we deduce about how large or small $|z-1|$ and $|z+i|$ could be? I tried drawing a diagram to get a feel but I don't know how to do anything more. I feel like $ 1< ...
3
votes
0answers
19 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
0
votes
0answers
46 views

How to find branch points for complex functions?

I'm looking for a standard way I can approach problems where I am tasked to find the branch points and branch cuts of a complex function. For instance, $$ f(z) = e^{(z^2+1)^{1/2}}$$ or $$ f(z) = ...
0
votes
0answers
46 views

Is there a name for the two parts of a complex number?

A complex number is the sum of a real number and an imaginary number. Is there a collective name for the two parts comprising a complex number, such that when used, it is (pretty) clear that the ...