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

learn more… | top users | synonyms

1
vote
0answers
14 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 = ...
1
vote
2answers
27 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.
-2
votes
3answers
54 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, ...
2
votes
0answers
20 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 ...
0
votes
3answers
55 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 ...
-3
votes
2answers
67 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 ...
0
votes
1answer
33 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 ...
1
vote
3answers
45 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 ...
0
votes
2answers
36 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 ...
1
vote
0answers
37 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 + ...
-1
votes
2answers
18 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 $ ...
0
votes
0answers
19 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 ...
0
votes
0answers
23 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}}$ $ ...
1
vote
1answer
35 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 ...
0
votes
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
votes
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 ...
2
votes
5answers
81 views

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

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

Compute all possible values of log(-j)

How do I find all possible values of $\log(-j)$? I need to use the equation
5
votes
1answer
63 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.
0
votes
1answer
40 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
votes
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
vote
4answers
39 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 ...
0
votes
1answer
29 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. ...
-3
votes
2answers
41 views

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

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
votes
6answers
68 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
votes
1answer
45 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
vote
2answers
34 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 ...
0
votes
1answer
38 views

Trigonometry question using complex numbers on the complex plane

I am not quite sure what this is asking, I tried to square these numbers and then convert into radians but it was not right. I am only used to graphing the absolute value of complex numbers. Let ...
-1
votes
0answers
38 views

Challenge: Rotation by 1 radian [on hold]

Prove in the most geometrical language you can (no Taylor series or pure algebraic manipulations) that $e^i$ represents a rotation by $1$ radian. Resorting to Euler's formula and identity are not ...
0
votes
1answer
28 views

Multiplication of two factors with complex numbers

I have the following to multiply ; $$(z-p-qi+\sqrt{t+ui})(z-p+qi+\sqrt{t-ui})$$ Now, I think that the product must not have any complex numbers... But here is what I get ...
1
vote
1answer
23 views

Evaluating complex functions integrals over closed curves

I recently evaluated the following two integrals: $\int_\gamma \dfrac{\bar z\,dz}{2i}$ where $\gamma$ is a circle with radius $r$ around some point. $\int_\gamma \dfrac{\bar z\,dz}{2i}$ where ...
0
votes
2answers
39 views

Argument for $(a+bi)^2$

I found out the modulus for $(a+bi)^2$, which is $$a^2+b^2$$ but I am unable to find the argument. I found out that $$\theta = \frac{2ab}{(a-b)(a+b)}$$ I don't know how to simplify further! Please ...
0
votes
3answers
22 views

Loci of Complex Equation

How does the loci of the equation $|z-(i+1)| = |1 + i|$ look like? I can't seem to visualise any points on the complex plane satisfying the above except the 2 obvious ones (2,2) and (0,0)... Is that ...
2
votes
1answer
27 views

If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then either $f\equiv0$ on $\Bbb C$ or $f(z)\not =0$ for all $z\in \mathbb C$.

Let , $f,g:\mathbb C\to \mathbb C$ be analytic such that $g(z)\not =0,\forall z\in \mathbb C$. If $\lim_{|z|\to \infty}\frac{f(z)}{g(z)}$ exists then prove that either $f\equiv0$ on $\Bbb C$ or ...
2
votes
1answer
39 views

Is there any interpretation to the imaginary component obtained when computing the geometric mean of a series of negative returns?

When computing returns in finance geometric means are used because the return time series of a financial asset is a geometric series: $\mu_r = \sqrt[T]{\prod_{t=1}^T r_t}$ where the return is computed ...
0
votes
2answers
30 views

Where does this equality come from? complex numbers rewritten

http://mathfaculty.fullerton.edu/mathews/c2003/ComplexSequenceSeriesMod.html See example 4.2. in above. They have $z_n = (1+i)^n$ and then they've rewritten that to a familiar $a_n+ib_n$ form ... ...
0
votes
1answer
13 views

Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
2
votes
1answer
44 views

Solve the complex equations

I have a question from complex calculus. How to solve this two equations: a) $$ \sin(z)=2015 $$ I know that $\sin(z)$ equals to $$ \frac{e^{iz}-e^{-iz}}{2i} $$ And i don't know whats next. b) $$ ...
-3
votes
1answer
36 views

Prove that $x^3+x^2+x$ is a factor of $(x+1)^n-x^n-1$ using complex numbers [on hold]

This question is given in my book under the complex number chapter but I can't understand how to solve it using complex numbers. It is given that $n$ is an odd integer greater than 3, but $n$ is ...
3
votes
6answers
370 views

Expansion of complex equation.

Find the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than ...
2
votes
5answers
55 views

Solving $\cos z = i$ for $z$

Solve $\cos z = i$ for $z$. What I have tried: $$\cos z = i$$ $$\frac{e^{-zi}+e^{zi}}{2}=i$$ $$e^{-zi}+e^{zi}=2i=2e^{\frac\pi 2 + 2\pi k},\quad k\in \Bbb Z$$ I would take logs, but then I would ...
7
votes
5answers
175 views

Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?

I am an $8^{th}$ grader that is taking Algebra I. But nearly everyday I try to learn things outside of what I am learning in class. Quite a while ago I discovered that $e^{i\pi} = -1$. This ...
0
votes
0answers
24 views

Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
0
votes
0answers
18 views

Harmonic Function Cauchy implication

Let $b$ be harmonic real valued on unit disk. Then I wish to prove that $\int_\alpha b =0$. I know that there exists $f$ holomorphic such that $\Re(f)=b$, and I know from Cauchys result that ...
0
votes
2answers
26 views

solving the limit $\lim_{n\to \infty}\sum_{k=1}^n|e^{(2πik)/n}-e^{(2πi(k-1))/n}$|

$$\lim_{n\to \infty}\sum_{k=1}^n\left|e^{(2πik)/n}-e^{(2πi(k-1))/n}\right|$$ i can solve it geometrically. but is there any way to solve it using Euler's formula ?, the answer will be one of these ...
1
vote
1answer
26 views

Schwarz Lemma, an onto map with $f'(0)>0$ is the identity

Let $f$ be $1-1$ holomorphic on unit disk onto itself. It satisfies (a) $f(0)=0$, (b) $f'(0)>0$. We need to prove that $f(z)$ is equal to $z$. I am stuck here, because I can prove using Shcwarz ...
0
votes
1answer
37 views

Unique properties of pure Imaginary numbers?

Are there any non trivial properties unique of the imaginary numbers? By trivial I mean stuff like $\bar a=-a$.
-1
votes
2answers
94 views

The real part of the sum $(i-1)+(i-1)^2+(i-1)^3…+(i-1)^{2013}$?

I'm not sure how to go around this one. Factorizing doesn't seem to work and there isn't a clear pattern to work by that I see. EDIT: So apparently I need to add context and stuff. I removed the ...
0
votes
1answer
22 views

Prove that if $Re(z)>0$ then $|z+\sqrt{z^2-1}| \ge 1$

This is probably a very basic question in complex numbers. First define $\sqrt{w} := \sqrt{|w|}e^{i(Arg(w)/2)}$ where Arg is the principal argument function. Prove that if $Re(z)>0$ then ...
0
votes
0answers
31 views

Complex Geometry Problem

Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Find [$PA_1^2 + PA_2^2 + \dots ...