Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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1answer
13 views

Proving for $w \in \Bbb C$ with modulus $1$ and argument $2 \theta$ that $\frac{w-1}{w+1}=i\tan \theta$

The complex number w has modulus $1$ and argument 2$\theta$ radians. Show that $$\frac{w-1}{w+1}=i\tan \theta.$$ Attempted solution: I just assumed that $w=1(\cos 2\theta +i \sin 2\theta)$ and ...
1
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1answer
9 views

Exponent identities with imaginary exponents ((a^i)^i)

I've been trying to understand how imaginary exponents work, and I think I mostly understand it, but I'm confused by something like $\left(a^i\right)^i$ (where $a$ is real). According to the normal ...
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2answers
22 views

Finding the sum of the trigonometric serie:

There are two series: $$1) 1+\dfrac{\cos{x}}{p}+\dfrac{\cos{2x}}{p^2}+...+\dfrac{\cos{nx}}{p^n}=\sum_{k=0}^{n}{\dfrac{\cos{kx}}{p^k}}$$ $$2) ...
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3answers
35 views

Finding the minimum value of $|a+b\omega+c\omega^2|$ if $a,b,c$ are unequal integers where $\omega^3=1$

My try 1: $$|a+b\omega+c\omega^2|\le\sqrt{|a+b+c||\underbrace{1+\omega+\omega^2}_0|}$$ Cauchy-Scwartz won't give us an upper bound since $a,b,c$ are nonequal integers. My try 2: ...
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2answers
40 views

divisibility of complex numbers

I want to show that $(a + bi)|(c + di)$ is equivalent to the statement that the ordinary integers $(a^2 + b^2)|(ac + bd)$ and $(a^2 + b^2)|(-ad + bc)$. I also want to show that $(a + bi)|(c + di) ...
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0answers
23 views

continuity of the complex square root function

I want to show that there is no continuous square root function in the complex plane, i.e. a function $f:\mathbb{C}\rightarrow\mathbb{C}$ with $f(w)^2=w$ for all $w \in \mathbb{C}$. I already ...
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2answers
874 views

Finding the square root of a complex number - why two solutions instead of four?

I want to find the square roots of a complex number, $w = a+ib \in \mathbb{C}$, i.e. I'm looking for solutions, $z = x + iy$, for the equation $z^2 = w$. This question has been asked here a couple of ...
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0answers
5 views

Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic number $\beta$ Pisot-like if $|\beta| > 1$ and all its conjugates lie inside the complex unit circle (here $|\cdot|$ is the usual absolute ...
2
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1answer
27 views

For what complex values of $z$ does the series $\sum_{n=0}^\infty \frac{z^n}{\log(n)}$ converge or diverge?

I used the ratio test to find that it converges when $|z| < 1$ and diverges when $|z| > 1$, but I'm not sure how to proceed with the $|z| = 1$ case. Because $z^n$ is function that traces out the ...
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2answers
21 views

Quadratic using the roots of unity, where $\omega^7 = 1, \omega \neq 1$

Say that $\omega$ is a complex number, where $\omega^7 = 1, \omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. $\alpha$ and $\beta$ are roots ...
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4answers
26 views

finding roots of cubic equation and the values of constants [on hold]

$x^3+px^2+qx+30=0$ where $p$ and $q$ $\in R$, has a root $1+2i$. $1)$ Find the other non-real root. $2)$ Find the third root of the equation. Hence, or otherwise, find the values of $p$ and $q$.
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3answers
50 views

How can I solve the simultaneous equations that arise in solving $\cos(z)=2$.

If I have $\cos(z)=2$ I can say $\cos(a+ib)=2$ using double angle ideas $\cos(a)\cos(ib)+\sin(a)\sin(ib)=2$ using Euler's formula $\cos(a)\cosh(b)+i\sin(a)\sinh(b)=2$ equating real and imaginary ...
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1answer
34 views

Solving $x^3 = -1$ for complex numbers [duplicate]

How can I solve for the complex solutions of $$ x^3 = -1 $$
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1answer
18 views

$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$

$$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$$ Now I am not sure how to prove this. Can I ignore the pesky square and do this? $$\lim_{z\to\infty} ...
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1answer
25 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
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1answer
52 views

The value of $1+2\alpha+3\alpha^{2}+…+n\alpha^{n-1}$ for complex $\alpha$

Compute the value of $$1+2\alpha+3\alpha^{2}+...+n\alpha^{n-1}$$ in the form of a complex number where $\alpha$ is a non-real complex $n^{th}$ root of unity. The answer given is : ...
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0answers
30 views

What's an elegant expression for a general conic using complex numbers?

A 'nice' form of writing a line through $2$ points $z_1,z_2\in\mathbb{C}$ is $$z\overline{z_1-z_2}+z_1\overline{z_2-z}+z_2\overline{z-z_1}=0$$ Now I'm asked to give a similar expression for a general ...
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2answers
19 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
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2answers
40 views

What do you call the subset of all the Gaussian integers having the smallest magnitude given an argument?

How is called the subset of Gaussian integers such that from all Gaussian integers having the same argument only one with the smallest absolute value is included? Is there a special name for them? ...
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2answers
49 views

Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.

Let , $f$ be entire function such that $|f\left(\frac{1}{n}\right)|\le \frac{1}{n^{3/2}}$ for all $n\in \mathbb N$. Then prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded. From the ...
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0answers
12 views

“Permutation” of squared norm and sum

In Problems and Solutions in Mathematics, 2nd edition, by Ta-Tsien, exercice 4312. Let $f$ be a periodic function on $\mathbb{R}$ with period $2 \pi$ such that $f|_{[0, 2 \pi]}$ belongs to $L^2(0, 2 ...
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1answer
46 views

Polynomial Equation Solution

Use Demoivre's theorem to show: $cos 7θ = 64 cos7 θ − 112 cos5 θ + 56 cos3 θ − 7 cos θ$ Hence,solve: $128x^7 −224x^5 +112x^3 −14x+1=0$ I've shown the first part and multiplied the equation by 2 and ...
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1answer
37 views

What is the number of complex integers inside a circle of radius r? [on hold]

What is the number of such complex integers, $z$, that $|z|\le r$? I am interested in a closed-form formula for integer $r$.
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3answers
51 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...
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2answers
31 views

Expansion of imaginary numbers

If $(1+i)^{100}$ is expanded, what is the value of the real part of the result? I know that this has to do with binomial theory and Pascal's triangle, but I don't know how to use it here.
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3answers
59 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
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0answers
25 views

Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
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3answers
63 views

Simplifying a Complex Number

I have $\left ( \frac{e^{i\frac{\pi}{3}}}{1+i}\right )^{2014}$. I wish to simplify this to standard form. I simplify to $\left ( e^{i\frac{\pi}{12}} \right )^{2014}$ I can evaluate and simplify ...
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2answers
72 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [on hold]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
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2answers
42 views

Prove that $\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$

Let $a,b,c$ be complex numbers such that $|a+b|=m$ and $|a-b|=n$ and $mn\ne0$. Prove that $$\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$$ I have tried using formula ...
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3answers
47 views

Is a matrix with complex entries invertable?

This is merely a question of interest and not for something I am doing in school. I have never seen a matrix with complex entries in class before, but mind you it was a limited linear algebra class, I ...
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2answers
37 views

Cauchy- riemann equations

Let $f(z) = u(x,y) + iv(x,y)$ be a complex function that is differentiable at the point $z_0 =x_0 + iy_0$. Prove that $f'(z_0)= \frac{\partial u}{\partial x} (x_0,y_0) + i \frac{\partial ...
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0answers
38 views

How to use the for re^itheta to prove this?

Can someone please explain how to use the form $re^{i\theta}$ and de Moivre's to prove that: $$\sum_{n=1}^N \frac{\sin n\theta}{2^n} = \frac{2^{N+1} \sin \theta + \sin N\theta - 2\sin(N + ...
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2answers
19 views

Representing a transformation from C to C with respect to the basis 1, i

I am having trouble understanding why the transformation: $ T(z) = (3+4i)z$ from C to C can be represented by the matrix $ \begin{bmatrix} 3, -4 \\ 4, 3 \end{bmatrix}$ with respect to the basis $ ...
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0answers
21 views

Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...
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0answers
25 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
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1answer
37 views

Prove that if $z$ is good then so is $z + r$ for every $r \in R$.

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ Note that $R$ is an integral ...
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0answers
14 views

An example of length, area or volume expressed as a complex number?

I sometimes have conversations with my fellow high school students about complex numbers and the existance of these "imaginairy" structures. I will then define the complex number $i$ algebraicly to be ...
2
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1answer
17 views

Use de Moivre's theorem to obtain an expression for $\sin^6x$ as a sum of terms in the form $\cos ax$

I'm not exactly sure if I'm on the right lines but I've started with a binomial expansion: $(\cos x+i\sin x)^6=\cos 6x +i \sin 6x= \cos^6 x + i(6\cos^5x \sin x)-15\cos^4x \sin^2x-i(20\cos^3x \sin^3 ...
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5answers
83 views

Argument of $z = 1 - e^{it}$

Let $t\in(0,2\pi)$. How can I find the argument of $z = 1 - e^{it}= 1 - \cos(t) - i\sin(t)$?
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1answer
26 views

Compute all possible values of log(-j)

How do I find all possible values of $\log(-j)$? I need to use the equation
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1answer
64 views

A small complex number whose total distance from other given complex numbers is large

Let $z_1,z_2,...,z_n$ be distinct complex numbers such that $|z_i|\leq1$. Is it true that there exists $z, |z|\leq1$ such that $\displaystyle\sum_{i=1}^n |z-z_i|\geq n$ ? Thank you.
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1answer
52 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
2
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0answers
32 views

Why is e used for polar form of complex numbers? [duplicate]

This is a real basic question. Why is $e$ the base for polar form of complex numbers? In high school maths I learned that e is useful in derivatives etc. And it's conventional to use it for ...
1
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4answers
42 views

Simplify $\frac{(\cos \frac{π}{7}-i\sin\frac{π}{7})^3}{(\cos\frac{π}{7}+i\sin\frac{π}{7})^4}$

Simplify $$\frac{(\cos \frac{π}{7}-i\sin\frac{π}{7})^3}{(\cos\frac{π}{7}+i\sin\frac{π}{7})^4}$$ I used de Morvre's theorem to get to $$\frac{(\cos ...
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1answer
32 views

What operation is “$\oplus$” in Lounesto's introduction to Clifford Algebras

I'm reading Lounesto's CLifford Algebras and Spinors and on page 26 (also below) he states the following: \begin{align} C\mathcal{l}_2=\mathbb{R}\oplus\mathbb{R}^2\oplus\bigwedge^2\mathbb{R}^2. ...
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2answers
41 views

$Arg(z+1) = \frac{π}{6}$ and $Arg(z-1) = \frac{2π}{3}$ [closed]

I'm really stuck I need to find z when $$Arg(z+1) = \frac{π}{6}$$ and $$Arg(z-1) = \frac{2π}{3}$$ Please help!!!!
2
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6answers
69 views

If complex numbers can be represented as vectors, why can't we define $2$-dimensional vector division just as complex division?

If complex numbers can be represented as vectors, why can't we define $2$-dimensional vector division just as complex division? Is there any inconvenient/incompatibility to this?
2
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1answer
50 views

Jacobi Identities

Can anyone guide me how can I prove these two identities? a)$$\prod_{n=1}^{\infty}\frac{1-q^{2n}}{1-q^{2n-1}}=\sum^{\infty}_{n=1}q^{n(n+1)/2}$$ b) ...
1
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2answers
36 views

Prove equality of two numbers written in complex polar form.

Show that these two numbers are equal: $$ z_1=\frac{e^{\tfrac{2\pi i}{9}}-e^{\tfrac{5\pi i}{9}}}{1-e^{\tfrac{7\pi i}{9}}} $$ and $$z_2=\frac{e^{\tfrac{\pi i}{9}}-e^{\tfrac{3\pi ...