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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3
votes
2answers
190 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
65 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
votes
1answer
75 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?
3
votes
1answer
52 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
59 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
votes
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
52 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
votes
0answers
21 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
38 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
votes
0answers
24 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
36 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
272 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
31 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
35 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, $a$ ...
1
vote
0answers
29 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 ...
1
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
42 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
27 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
30 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
33 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
91 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
25 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
37 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
188 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
44 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 ...
4
votes
0answers
28 views

Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
2
votes
1answer
55 views

What do $\int_{-1}^1\frac{dx}{2x+1-2i}$ and $\frac12\log(2x+1-2i)$ mean?

Suppose we want to evaluate $$I=\oint_C\frac{dz}{z+\frac12}$$ where $C$ is the unit square with diagonal corners at $-1-i$ and $1+i$. If we let $z:=re^{it}-\frac12$, then ...
22
votes
3answers
2k views

Can I compare real and complex numbers?

I'm calculating the eigenvalues of the matrix $\begin{pmatrix} 2 &0 &0& 1\\ 0 &1& 0& 1\\ 0 &0& 3& 1\\ -1 &0 &0 &1\end{pmatrix}$, which ...
-3
votes
1answer
38 views

Formula Derivation

What is the formula that can be derived using the values and formulas below to get the value of K43? Here's what I have, so far, but no luck in getting the correct formula: 0 = 925191 - 119355 - ...
1
vote
1answer
28 views

An inequality for complex number $|a+b|^p \sim |a|^p+|b|^p$.

I know that for any nonnegative numbers $a,b$ and $1\leq p<\infty$ then $a^p+b^p\leq (a+b)^p\leq 2^{p-1}(a^p+b^p)$. Now we need to find the similar inequalities for complex numbers. My question ...
2
votes
3answers
157 views

$(-1)^{\sqrt{2}} = ? $

This popped up when I was thinking about $$(-1)^{\frac {p}{q}}$$ where $ p $ and $q$ are integers such that $\gcd (p,q) = 1$ If $p$ is even : $(-1)^{\frac {p}{q}} = +1$ If $q$ is even : ...
2
votes
2answers
50 views

Prove that $\max_{|z| = 1} |P(z)| \ge 1$

I got stuck on this problem: Given a polynomial on complex plane $P(z) = z^n + a_{n-1}z^{n-1} + ... + a_1 z + a_0$ for $z \in \mathbb{C}$. Prove that $\max_{|z| = 1} |P(z)| \ge 1$ What I tried ...
0
votes
3answers
28 views

Complex Numbers and Euler/Polar Form

Say you have a complex number with $|z|=2$ and argument of $-\pi/3$. Why is it not valid to say $e^{-\pi/3i} = e^{5\pi/3i}$? Is it still valid to say $2cis(-\pi/3) = 2cis(5\pi/3)$?
0
votes
1answer
34 views

Powers with complex/negative bases

If x can be a positive real number (for example a fraction with a numerator and denominator), then why does the following relationship hold true only if and only if a and b are strictly positive real ...
0
votes
3answers
38 views

determinant of SU(3) matrix

I don't understand the determinant condition on SU(3) group, broadly. I know that the determinant of such matrices should be equal to 1. But what is the real intention of that 1? Is it the real ...
0
votes
3answers
81 views

Solutions of $z^5 = -1$

I have found the solutions to $z^5=-1$ but I have to use the following factorization to find the complex number produced when all solutions are multiplied. Each solution is denoted by $z_0-z_4$: ...
0
votes
1answer
72 views

Are all numbers expressible as a complex number? [closed]

Are there any numbers that are not elements of the complex field? Follow-up questions: Are p-edic fields subfields of the complex field? Can quaternions be viewed as a complex vector space in three ...
0
votes
2answers
59 views

Finding the 5th root of a complex number

I've got myself really confused on this question, and I'm not sure where to go from my attempt. Thanks for any help! QUESTION: Find the 5th roots of $1 + 2i$. ATTEMPT: I've began by finding the ...
0
votes
3answers
146 views

$\sqrt{x}=-1$. How can I solve it?

I am so curious about this equation: $\sqrt{x}=-1$ Does the $x$ where $x\in \mathbb{C}$ exist? How can I solve it?
2
votes
1answer
107 views

Showing $3<\pi<2\sqrt{3}$ using complex analysis

First of all we define $\pi$ to be $\pi=2\sup\{t\geq 0 :\; \text{for}\; 0\leq s \leq t, \Re(e^{is})\geq 0 \; \text{and}\; \Im(e^{is})\geq 0\}$ And we know that $e^{\frac{\pi}{2}i}=i$,$e^{\pi i}=-1$ ...
0
votes
1answer
11 views

Are G-numbers equivalent to Eisenstein integers?

In 100 Great Problems of Elementary Mathematics the author terms the set of numbers $$xO+yJ:x,y\in\mathbb{Z}$$ where $J=-\omega^2$, $O=-\omega$ and $\omega=e^{2\pi i/3}$ as G-numbers. These are used ...
1
vote
0answers
23 views

Relation between the Complex Number System and Vector Spaces

Given that any Complex Number $z$ can be represented as a Vector in $\mathbb{R^2}$ and since a Vector is nothing more than an element of a Vector Space (in its most general form), does that not imply ...
0
votes
1answer
48 views

Hyperbolic Trigonometry with Complex Numbers

I was trying to show the following complex hyperbolic trigeometric relation ...