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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2
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0answers
34 views

$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
-2
votes
1answer
32 views

Help with complex numbers question [closed]

Could someone please solve and explain this question about complex numbers? I'm having a lot of difficult understanding parts (b)(iii) and (b)(iv) especially. I understand what they are asking but ...
1
vote
2answers
39 views

Prove a doubly periodic entire analytic function in complex plane is a constant [duplicate]

I got stuck on this problem. So I really appreciate if anyone can give me some hint to move on. Thanks a lot. Prove that an entire analytic function $f:\mathbb{C} \rightarrow \mathbb{C}$ is a ...
1
vote
2answers
53 views

Complex Analysis: How isolated singular points behave

I am working on the following question: Suppose $z_0 \in \mathbb{C}$ is an isolated singular point of the function f of a given type (removable, pole of order N, essential). Show that $z_0$ is an ...
0
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0answers
28 views

What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $$p=1+\mathtt{i}=\bar{q},\ \ q=1-\mathtt{i}=\bar{p}.$$ Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & ...
0
votes
1answer
45 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
0
votes
1answer
37 views

the closure of the set $\{ e^{in\theta}:n\;\text{non-negative integer numbers} \}$

May I ask a question about the closure of the set $\{ e^{in\theta}:n\;\text{non-negative integer numbers} \}$, where $\theta\in\mathbb R$. Many thanks.
2
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3answers
212 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $z_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the ...
0
votes
1answer
21 views

Find all $z = (x, y)$ such that $z^2 + z + 1 = 0$

Find all $z = (x, y)$ such that $z^2 + z + 1 = 0$ I just started doing complex numbers and am unsure how to solve this problem.
0
votes
1answer
40 views

Complex exponential with 2 pi

I wonder why is it wrong to do the following: $e^{i2\pi x}=(e^{i2\pi})^x=1^x=1$ for a real $x$ but not for an integer $x$
0
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1answer
32 views

Explanation of i to the i power? [duplicate]

Could somebody give me a good explanation for how $i^i$ works? I'm a junior and just now getting to this. I'm also too hard pressed for time to dive into exploring it myself.
1
vote
3answers
62 views

How do I compute the following complex number? [on hold]

This was the problem I was given: Compute the complex number for $\frac{(18-7i)}{(12-5i)}$. I was told to write this in the form of $a+bi$. So please give me a hint of how to do this. :)
2
votes
2answers
53 views

prove $\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$

Today I found the identity : $$\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$$. How to prove or disprove this? Thank you.
0
votes
2answers
40 views

Fiding imaginary part of a complex number [closed]

What is the imaginary part of $i^i$ ? I've tried multiple approaches, including using log. I can't seem to understand how to work with complex numbers as logarithmic functions. Also, it would ...
0
votes
0answers
12 views

Equality involving holomorphic function and its series coefficients [duplicate]

Function $f(z)=a_0 + a_1z +a_2z^2+...$ convergences on $\left\{z:|z|<R\right\}$. Prove that for any $0<r<R$ $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^{2}dt= ...
0
votes
0answers
18 views

$z$ satisfies the relation $|z-(\alpha^2-7 \alpha+11 -i)|=1$ and $\alpha \in R$. [closed]

If $z$ satisfies the relation $|z-(\alpha^2-7 \alpha+11 -i)|=1$ and $\alpha \in R$. Also $argument(z) \geq \frac{\pi}{2}$ is satisfied by at least $z$. Then answer the following question $1$. The ...
1
vote
1answer
38 views

Computation of an inverse trigonometric series using complex numbers

The following is a popular question (in competitive exams) in India: Compute the value of $S=\displaystyle \sum_{k=1}^{\infty} \tan^{-1}\left( \dfrac1{2k^2}\right)$. I can compute the value by ...
3
votes
2answers
166 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
2
votes
3answers
64 views

$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $

Let $\omega \ $be a root of the polynomial $\ x^{2016} +x^{2015}+x^{2014}+...+x+1=0 \ $. Then find the value of the following sum: $$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $$ Well I have simplified ...
0
votes
1answer
14 views

Euler formula - equivalent angles

How does $e^{−5π i/6} = − \cos(π/6)−i\sin(π/6)$? Shouldn't the argument for the $\cos$ and $\sin$ be $5*\pi/6$? Thanks
0
votes
0answers
25 views

Solution using complex numbers

A ray of light is travelling along $\mathbf{i}+\sqrt{3} \, \mathbf{j}$, it hits a plane mirror and is reflected along $\mathbf{i}-\sqrt{3} \, \mathbf{j}$. What is the angle between normal and the ...
0
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1answer
60 views

What is solution of $j^3$ (j is complex number)?

I have a confused with this problem? I calculate this by 2 ways: $$j^3 = jj^2 = j(-1) = -j$$ $$j^3 = j^{\frac{12}{4}} = (j^{12})^{0.25} = 1^{0.25} = 1$$ Why does it have different result?
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votes
0answers
19 views

bessel function integration [closed]

$$ I(u,v_p) = \int_{0}^{{\it v_p}}\! \left| \int_{0}^{1}\!{{\rm e}^{iu{\rho}^ {2}}}{{\rm J}_{0}\left(v\rho\right)}\rho\,{\rm d}\rho \right| ^{2}v\,{\rm d}v $$ Suppose: P=1 and J is the ...
3
votes
1answer
45 views

What is $\frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2$?

Consider the function $$\Bbb C-\{0\}\to\Bbb R,\quad z\mapsto\log|z|^2.$$ What is $$\frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2?$$ Try: I am no sure if the second step is justified, but ...
0
votes
1answer
53 views

Complex Numbers— A Different World [duplicate]

I know complex numbers but what is the meaning of "complex" in "Complex Number"? Does "complex" mean "complicated here? Are complex numbers used in easy real world problems? Give me examples please.
0
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1answer
23 views

Rearrangement of Complex Sin and Cos

From my complex numbers course notes, there is the following derivation: The definitions of sin and cos I'm very comfortable with, but I cannot see how we get from the definition to the given ...
0
votes
1answer
49 views

Euler's Formula - Complex Numbers

How does one get from $$\frac{2\pi\mathrm{i}}{6}\left(\mathrm{e}^{−5\mathrm{i}\pi/6}+\mathrm{e}^{−5i\pi/2}+\mathrm{e}^{−\mathrm{i}\pi/6}\right)$$ to ...
2
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0answers
19 views

Can you find an interpretation of the following arithmetical function?

For integers $n\geq 1$, taking $k\geq 1$ for $$z_k:=\mu(k)+i,$$ where $\mu(k)$ is the Möbius function and $i=\sqrt{-1}$ the complex imaginay unit, then we define the (real) arithmetical function ...
1
vote
1answer
34 views

About the definition of isolated singularity of a complex function

I'm learning the part Isolated Singularity Categorization, and there's a point in the definition of the isolated singularity which confused me a lot: A function $f$ has an isolated singularity at ...
0
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0answers
23 views

I need help solving the following Euler Relations

I need to determine the smallest positive value of $\alpha$, $\beta$,and $\gamma$ that satisfy that following relations: $$e^{i \alpha }=i$$ $$e^{i \beta}=-1$$ $$e^{i \gamma}=-i$$ I know that I need ...
0
votes
0answers
13 views

Find the maximum value $a$ such that if $|z-5-6i| \geq a$, then $|z+3| \leq 1 $ for at least one $z$.

Question: Find the maximum value $a$ such that if $|z-5-6i| \geq a$, then $|z+3| \leq 1 $ for at least one $z$. What I have done Considering $$|z-5-6i| \geq a$$ $$|x+iy-5-6i| \geq ...
0
votes
1answer
31 views

Geometric Proof of DeMoivre's Formula

I want to know if there were a geometric proof of DeMoivre's formula. My attempt was starting with an easy complex number and observing patterns, then generalizing that pattern. If you start with ...
2
votes
2answers
22 views

Understanding simplifications of complex terms $\exp(-ik\pi/4)$

I read that $1\over{2}$$\pi$$i$($\exp[-3i\pi/4]+\exp[-9i\pi/4])$ = $1\over{2}$$\pi$$i$($-\exp[i\pi/4]+\exp[-i\pi/4])$ = $\pi$$\sin(\pi/4)$ = $\pi\sqrt{2}$ Can you help me to understand how we move ...
5
votes
4answers
259 views

Easy partial fraction decomposition with complex numbers

There is an easy method to perform a partial fraction decomposition - described here, under the "Repeated Real Roots" title, for the coefficient A2. The problem is ...
0
votes
1answer
26 views

Radius of convergence of two series [duplicate]

An unproven proposition in my book states that if the series of $a_{n}z^n$ has radius of convergence $R_1$ and the series of $b_{n}z^n$ has radius $R_2$. Then the radius of convergence of ...
0
votes
3answers
29 views

Sum of n-th roots of unity [duplicate]

I'm being asked to prove that $$1 + \omega + \omega^2 + ... + \omega^{n-1} = 0$$ where $\omega \ne 1$ is an n-th root of unity, and I don't know where to start I feel like there's something terribly ...
3
votes
2answers
47 views

Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$.

Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$. Prove that the sum $a_0 + a_1z + a_2z^2 + \cdots + a_nz^n +\cdots $ is ...
1
vote
1answer
66 views

Is the language of complex numbers regular?

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and i is the imaginary unit, that satisfies the equation $i^2 = −1$. In this expression, $a$ ...
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0answers
25 views

Determine the number of zeros using the Argument Principle

I'm tasked with finding the zeros of $f(z)=z^3+1$ that lie inside the first quadrant using the Argument Principle, which I have simplified below: $$N=\frac{1}{2\pi}[arg(f(z))]_C$$ where N represents ...
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vote
1answer
37 views

Laurent series for $z^{2} e^{1/z}$ at $z = \infty$

I just found the Laurent series for $z^{2}e^{1/z}$ for $z = 0$, and now I need to find it at $z = \infty$. (for $z=0$, it was $\displaystyle \sum_{n=0}^{\infty}\frac{z^{2-n}}{n!}$, by the way). I'm ...
0
votes
3answers
41 views

How to prove that $\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ [duplicate]

Prove that $\displaystyle\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ for $n\in\mathbb{N},n>1$ I'm thinking at a demonstration by induction, as base case $n=2$ ...
3
votes
1answer
26 views

Laurent expansion of $\frac{1}{(z-a)^{k}}$, $k \in \mathbb{N}$

I need to expand the function $f(z)=\frac{1}{(z-a)^{k}}$ where $a \in \mathbb{C}$, $a \neq 0$, $k \in \mathbb{Z}$, $k>0$ in a Laurent series in the annuli (a) $0< |z|<|a|$ (b) $|a|<|z|$ ...
2
votes
2answers
25 views

Expand the function $f(z)=\frac{1}{(z-a)(z-b)}$ where $0 < |a| < |b|$ in a Laurent series in different annuli

I have to expand the function $f(z) = \frac{1}{(z-a)(z-b)}$ where $a, b \in \mathbb{C}$, $0 < |a| < |b|$ in the following annuli: (a) $0<|z|<|a|$ (b) $|a|<|z|<|b|$ (c) ...
1
vote
2answers
29 views

What is the solution of $\sin z=\cosh 4$?

What is the solution of $\sin z=\cosh 4$? By putting $z=x+iy$ I managed to find that the real part of $z$ is $x= \frac \pi 2+2n\pi $, but the imaginary part is contradictory giving negative value of ...
0
votes
2answers
89 views

Help with complex numbers geometry proof

See this link. The last step is skipped, because it is claimed to be trivial, but apparently there is a gap in my knowledge. $M$ is $\frac{1}{2}(b+c)$ and $H$ is $\frac{1}{2}i(b+c)$, but how do you ...
2
votes
1answer
52 views

$n$ complex numbers with modulus $1$

The problem: Let $z_1$,$z_2$,...$z_n$ $(n \geq 3)$ be complex numbers such that $\left| z_1 \right|=\left| z_2 \right|=\ldots=\left| z_n \right|=1$. Then show that the following statements are ...
0
votes
3answers
24 views

Ring Extension: Mapping: $ \mathbb Q[\sqrt d] \rightarrow \mathbb Q$

Show that the Norm: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q, (r+s\sqrt d) (r-s\sqrt d) = r^2 - ds^2$ is multiplicative, i.d. $N(xy) = N(x)N(y)$ How to show it without computing? (I tried to do it ...
0
votes
1answer
34 views

Is the dimension $-1$ the real $0$th dimension and does this all make sense?

I know there are at least two questions on this site that ask about the negative dimensions. But I want to ask something more than that. We have a number line. It contains all the real numbers we can ...
6
votes
1answer
186 views

$m+ni+k\lambda,\,\Re(\lambda),\Im(\lambda)\notin \mathbb{Q}$ is dense in $\mathbb{C}$!

As said in the comments below, it's needed to suppose $\{1,\Re(\lambda),\Im(\lambda)\}$ linearly independent over $\mathbb{Q}$, otherwise the result is false, according to Christian's example. ...
1
vote
2answers
40 views

Determine the largest open set to which $f(z)=\sum_{n=1}^{\infty}(-1)^n(2n+1)z^{n}$ can be analytically continued

Let $U=B_1(0)$ and $$f:U \rightarrow \mathbb{C},\qquad f(z)=\sum_{n=1}^{\infty}(-1)^n(2n+1)z^{n}.$$ Determine the largest open set to which $f$ can be analytically continued Remark: I was given ...