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
21 views

For which values of $z \in \mathbb{C} $ do we have $ A_2 (z) = 0 $?

$ \forall z \in \mathbb{C} $ : $ e^{ jz} = \displaystyle \sum_{ n \geq 0 } \dfrac{ (jz)^{n} }{n!} = \displaystyle \sum_{ n \geq 0 } \dfrac{ x^{3n} }{ (3n)!} + j \displaystyle \sum_{ n \geq 0 } \dfrac{ ...
0
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1answer
24 views

Representing particular sets of complex numbers

I am supposed to represent the following sets: \begin{align}A&=\{z\in\mathbb{C}:\Re(2z+iz)<0<\Im(z^2)\}, \\ B&=\{w\in\mathbb{C}:w=z^2, z\in A\}, \\ C&=\{u\in\mathbb{C}:u=1/z, ...
1
vote
1answer
25 views

Maximum value of arg z

On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $|z-2-i|\leq1$ and $|z-i|\leq|z-2|$. Calculate the maximum value of arg $z$ for ...
3
votes
1answer
37 views

Why can we identify complex numbers as points on a plane?

Modern mathematicians seem to define the complex number $a+bi$ as the ordered pair $(a,b)$, with the usual rules for complex addition and multiplication. I'm reading a book on the history of the ...
0
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1answer
14 views

Find the residue of $f(z)=\frac{1-\cos z}{2\sin z-\sqrt{3}}$ at $\pi/3$

Find the residue of $f(z)=\frac{1-\cos z}{2\sin z-\sqrt{3}}$ at $\pi/3$. Here is my attempt: The Taylor series for $\cos z$ and $\sin z$ about $z=\pi/3$ are \begin{align*} \cos z &= ...
0
votes
0answers
12 views

Terminology for a Complex Semicircle

A while back I read about a special type of number system termed (if I remember correctly) as "degrees of sign". The idea was that numbers sat on a series of 0 to 180 degree rays. Positive ways the 0 ...
1
vote
1answer
23 views

How can I make sure that the classical way of calculating the characteristic function of an exponential holds?

Given $f(x)=\lambda e^{-\lambda x}$, I want to find $\phi(t) = E(e^{itx})$ (characteristic function). Classical way: \begin{align} \phi(t) &= \int_0^{\infty} e^{itx}\lambda e^{-\lambda x} dx \\ ...
0
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0answers
16 views

Taking a linear combination of complex numbers

This may be a very silly question, but I will ask it regardless and apologize later if necessary. Suppose $w$ and $z$ are complex numbers. When we say "take a linear combination of $w$ and $z$" does ...
0
votes
2answers
24 views

Complex inversion map

How do I show that the map $f: \mathbb{C}\setminus \{0\} \to \mathbb{C}\setminus \{0\} $. $f(z): z \mapsto \frac{1}{z} $ maps circles to either a circle or a line?
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2answers
28 views

Primitive 8th roots of unity in Z17

If $\omega=\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2}$, then $\omega$ is an 8th root of unity. And I know $\omega,\omega^3,\omega^5$,and $\omega^7$ are furthermore primitive 8th roots of unity in ...
0
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2answers
35 views

Points at which $f(z) = z^{2} \bar{z}$ is differentiable?

So I know that $f(z)$ is diff'able at some point $z_{o}$ if the limit of the following exists: $$ f'(z_{0})= \lim_{h\to0} \frac{f(z_{0}+h) - f(z_{0})}{h}.$$ In the case of $f(z) = z^{2} \bar{z}$, I ...
1
vote
1answer
44 views

Evaluate $\int_{\partial \mathbb{B}(-i,3)}\frac{\sin(z)}{(z-3)^3}\, \mathrm{\mathop{d}}z$ using Cauchy's Integral Formula

I would like to evaluate $$\int_{\partial \mathbb{B}(-i,3)}\frac{\sin(z)}{(z-3)^3}\mathrm{\mathop{d}}z$$ using Cauchy's Integral Formula. Notation: We denote $\partial \mathbb{B}(a, R)$ to be the ...
2
votes
2answers
37 views

What is represented by the set of complex numbers z satisfying the equation $(3+7i)z+(10-2i)\bar{z}+100=0$?

I have an objective type question :- The set complex numbers $z$ satisfying the equation:- $$(3+7i)z+(10-2i)\bar{z}+100=0$$ represents:- A) A straight line B) A pair of intersecting straight lines C) ...
1
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2answers
52 views

How to generalise this complex equation?

I am trying to generalise the statement for $n$ complex numbers: For any complex numbers $a,b,c$ with property $|a|=|b|=|c|=r\neq 0$. Prove $|\frac{ab+bc+ca}{a+b+c}|=r$ I proved this by showing ...
2
votes
1answer
511 views

Quadratic formula with complex coefficients

Let $a,b$ and $c$ be complex numbers. I'm trying to prove that this version of the usual quadratic formula: $$z=\frac{-b+(b^2-4ac)^{\frac{1}{2}}} {2a}$$ solves the quadratic equation ...
1
vote
1answer
88 views

Representing $ 2 i \sin(2 \pi n z) $ as a product

I'm currently reading a surprising proof of the Quadratic Reciprocity Law, which uses the following function: $ f(z) = 2 i \sin(2 \pi z) $ One of its properties is that $$\frac{f(nz)}{f(z)} = ...
0
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0answers
20 views

Quick De Moivre's Theorem Question/Example

Could these also be solutions since 4 is even? I am relatively new to complex bases and fractional powers.
2
votes
2answers
48 views

Derive identities for $\cos(4x)$ and $\sin(4x)$ using following fact

So I need to use the fact that: $$\cos(4x) + i\sin(4x) = \left(\cos(x) + i\sin(x)\right)^4$$ to derive identities for $\cos(4x)$ and $\sin(4x)$ in terms of $\cos(x)$ and $\sin(x)$. I'm not sure how to ...
2
votes
1answer
30 views

Convergence of $\sum\frac{(n-a)^2}{(n-b)^3}$ with $a,b$ complex numbers.

Find the complex constant $a, b$ for which $\sum\frac{(n-a)^2}{(n-b)^3}$ converges and diverges.
4
votes
1answer
21 views

Differential Equations Complex Eigenvalue functions

Show that a function of the form $x(t) = K_1 \cos\beta t + K_2 \sin\beta t$ Can be written as $x(t) = K\cos(Bt-\phi)$ Where $K = \sqrt {K_1^2 + K_2^2}$ I know that linear systems with complex ...
1
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0answers
56 views

Find imaginary part of complex expression

Given the system of ODEs, $$x'=x^3-3xy^2$$ $$y'=3x^2 y-y^3,$$ it can be shown that the system may be written as $z'=z^3$, where $z=x+iy$. However, I don't seem to get how to show that $\Im ...
1
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1answer
24 views

A deduction from $x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos{\frac{2k\pi}{n}}+1]$

Prove that $$x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos{\frac{2k\pi}{n}}+1]$$ if $n$ be an even positive integer. Hence deduce that ...
0
votes
1answer
18 views

In this example, why are the points exterior to the circle $|z|=1$ mapped onto the nonzero points interior to it?

On page $314$ it says "the points exterior to the circle $|z|=1$ mapped onto the nonzero points interior to it". Why/How is this so? From Complex Variables - Churchill/Brown
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2answers
14 views

what is the set of points in the complex plane which satisfies |z| = Re(z) + 2?

what is the set of points in the complex plane which satisfies |z| = Re(z) + 2? so $ \sqrt{x^2+y^2} = x + 2 $ this is not a circle or anything and it asks me to sketch it what should I do
1
vote
2answers
61 views

Is $ze^{z}$ differentiable?

As the title says, I am looking at the function: $$f(z) = z e^{z},$$ and I want to know whether it is differentiable - and if so, to find its derivative (which I can do). Usually I would either ...
0
votes
0answers
17 views

What other methods can you use find the roots of a complex equation without using de Moivre's Theorem?

e.g. $z^n=x+iy$ is solved by $z=r^\frac1n[\cos(2\pi k+\theta) + i\sin(2\pi k+\theta)]^\frac1n$, where $r = |z|$ and $\theta=arg(z)$ Can you find $z$ without using the above formula?
-1
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4answers
78 views

How in this world can I simplify this $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????

I have a problem, obviously. I am doing some maths and now I have to simplify this: $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????. But I just don´t know how ???? I´ve started simplifying by ...
0
votes
3answers
19 views

Solutions to the complex equation z^n=w with one solution given

In an old test paper $z_0=2+i$ is given as one solution to $z^4=-7+24i$ and we are asked to find further solutions. In the solution is given $$z_1=-z_0=-2-i$$ as $$z^4_1=(-1)^4z_0^4=z_0^4$$ I ...
0
votes
2answers
26 views

Condition on $k$ if $|z-z_1|^2+|z-z_2|^2=k$

How can we prove that $$|z-z_1|^2+|z-z_2|^2=k$$ will represent a circle if $|z_1-z_2|^2 \leq 2k$ Please give me some hints to initiate this question.
0
votes
0answers
23 views

Understanding complex multiplication as vector addition

I've been studying complex multiplication and vector math lately. Below is a visual representation of what happens when one multiplies a complex number $ 1 + xi $ by itself repeatedly, plotting the ...
0
votes
1answer
25 views

Plotting numbers in the complex plane using WolframAlpha

I read this article on understanding imaginary numbers as rotations of real numbers in the complex plane. Having read it, it's easy for me to see how the real number $1$ is simply a point on the real ...
6
votes
5answers
463 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
0
votes
2answers
62 views

The number of solutions for n raised to a complex exponent

My understanding is that there is one and only one solution when solving for $z$ when $z = n^s$, where $s$ is a complex number of the form $a + bi$. However, there are many solutions to $z$ when ...
0
votes
1answer
35 views

Evaluate $\ln[(1+i)^7], \mathrm{Re}[\cos(1+i)]$ and $|e^{3+i\pi/4}|$

I am unable to verify my answers or methods for the 3 parts of this question, any input would be appreciated. I have attached my solutions below.
1
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1answer
26 views

Proofing de Movire without Induction and in a neat way

The "usual way" gone for proving de Movire is via the road of induction. However this road get tiresome and thus wondered, if there were another way. However I came up with a proof that relies on ...
0
votes
1answer
14 views

Positivity for Hermitian dot product?

One of the requirements for the Hermitian dot product is positivity, i.e. $||v||^2 \ge 0, $ and $||v||^2 = 0 \iff v=0$. I was wondering what exactly this means in the complex numbers. Does it mean ...
0
votes
1answer
57 views

Explain this complex number simultaneous equation step. [closed]

The explanation appears on this web page: OPs problem question Following through, I see everything until the move linking these two steps: $15a - 10b = 7a - 6b$ $8a = 4b$ What mathematical logic ...
2
votes
1answer
99 views

Variation of argument of a complex function

Variation of Argument : Definition( Collect from my book ) : Let $f$ be analytic inside and on a sinple closed contour $C$ except possibly for poles inside $C$ and $f(z)\not=0$ on $C$. As $z$ ...
2
votes
2answers
143 views

Simplifying this (perhaps) real expression containing roots of unity

Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant. Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$. ...
0
votes
1answer
45 views

Convergence of series $\sum 2^n \sin\left(\frac{a}{3^n}\right)$ with $a$ complex numbers. [closed]

Find the complex numbers $a$ (if exist) for which is convergent the series $$\sum 2^n \sin\left(\frac{a}{3^n}\right)$$
3
votes
2answers
51 views

For three complex numbers we have $|z_1|=1$ ,$|z_2|=2$ ,$|z_3|=3$ and $|9z​_1z_2 + 4z_1z_3 + z_2z_3|=12$

For three complex numbers we have: $|z_1|=1$ ,$|z_2|=2$ ,$|z_3|=3$ and $|9z​_1z_2 + 4z_1z_3 + z_2z_3|=12$ Then find value of $|z_1 + z_2 + z_3|$ I took $z_1=1(\cos A+i\sin A),z_2=2(\cos ...
3
votes
1answer
74 views

Ratio distance similarity transformations: $|\varphi (a)- \varphi (b)| = k|a-b|$

I am struggling with the following group geometry question. I am given that: A simililarity transformation is a non-constant map $\varphi : \mathbb{R^2} \to \mathbb{R^2}$ that leaves the ratios of ...
1
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3answers
36 views

$|z_1+z_2|>|z_1-z_2|$ implies $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$

For two complex numbers $z_1$ and $z_2$, it is given that: $$|z_1+z_2|>|z_1-z_2|$$ How could we prove that $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$ If I take ...
0
votes
2answers
172 views

Proof of index laws for complex numbers

Can someone give a proof that index laws (and hence log laws) apply for complex numbers in the same way they do to reals, specifically that: $(a^{ix})^n = a^{ixn}$ Assuming $a, x, n$ are real and ...
1
vote
2answers
24 views

Find all the points where $f$ is analytic with $f(z)= \frac{z^2+1}{(3z-1)(z-i+1)}$.

I start by expanding the denominator and separating the real and imaginary but get stuck when deciding what my $u$ and $v$ should be. Thanks.
1
vote
1answer
29 views

How to prove the following for an inner product space?

How to prove $\langle x,y\rangle=ax_1\bar{y}_1+bx_2\bar{y}_2$ defines an inner product of $\Bbb{C^2}\iff$$a,b\in\Bbb{R^+}$ $\Rightarrow$ if $\langle x,y\rangle=ax_1\bar{y}_1+bx_2\bar{y}_2$ defines an ...
0
votes
2answers
63 views

How to solve $\textrm i^y=y$? [duplicate]

What is the value of $\textrm i^{{\textrm i}^{\textrm i \dots}}$? My try: put $\textrm i^{\textrm i ^{\textrm i \dots}}=y$, so we get $\textrm i^y=y$. Now how to solve this equation?
0
votes
1answer
45 views

Radius of convergence of $\sum\limits_{n}c_{n}z^{n^{2}}$ given that the radius of convergence of $\sum\limits_{n}c_{n}z^{n}$ is finite and nonzero

I know that the radius of convergence of a given power series $\sum_{n=1}^{\infty}c_{n}z^{n}$ is $R$, where $0<R<\infty$. Given this information, I need to find the radius of convergence of ...
4
votes
1answer
98 views

Where are the points of $\{z^{z}\in \mathbb{R}|z\in\mathbb{C}\}$?

I was curious to find out when, given $z_1,z_2\in \mathbb{C}$, I have that $z_1^{z_2}\in \mathbb{R}$ or $z_1^{z_2}\in \mathbb{I}$, where $\mathbb{I}$ is the set of imaginary numbers. So I used the ...
1
vote
4answers
41 views

Evaluate $\Re[\cos(1+i)]$

Evaluate $\Re[\cos(1+i)]$. The trigonometric function in the expression is throwing me in a loop and need some guidance on how to evaluate this. Thanks.