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
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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
39 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
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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
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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
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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
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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
57 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?
-1
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0answers
17 views

bessel function integration [on hold]

$$ I(u,v_p) = \int_{0}^{{\it v_p}}\! \left| \int_{0}^{1}\!{{\rm e}^{iu{\rho}^ {2}}}{{\rm J}_{0}\left(v\rho\right)}\rho\,{\rm d}\rho \right| ^{2}v\,{\rm d}v $$ 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
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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
votes
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
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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
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4answers
258 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
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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, $a$ ...
1
vote
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 ...
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
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
183 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 ...
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
51 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
26 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
151 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
45 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
27 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
30 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
33 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 ...