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

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1answer
49 views

Complex Numbers and Circle

I am having a problem with the following question. They ask me for what values of a+bi is $e^{(a+ib)t}$ a circle. We have $t \in \mathbb{R}$ I think that since the modulus of $e^{(a+ib)t}$ is ...
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2answers
58 views

Set of points that fulfill a formula

Suppose $u=\frac{z(1+i)-i}{z+1}$ as $z\in \mathbb{C} \setminus\{-1\}$ What is the set of points $M(z)$ for which $u$ is a real number? What is the set of points $M(z)$ for which $u$ is pure ...
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14answers
2k views

How do I understand $e^i$ which is so common?

Raising something to an imaginary number is weird, I have a hard time wrapping my head around that. And e seems even more common and comes up in many situations, such as: the non-geometric ...
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3answers
95 views

A set such that for every $i \in S$, $i^2 \in S$ as well.

I was considering the largest possible set of complex numbers which contained the squares of every element; that is, the largest possible set $S$ such that for every $i \in S$, $i^2 \in S$ as well. ...
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0answers
58 views

Distances on $ C_\infty $

Let $ C_\infty $ be the extended complex plane and let $ dist_{S^2} $ be the metric induced on $ C_\infty $ by the stereographic projection ($ dist_{S^2} $ is simply the standard distance function ...
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2answers
104 views

equality between modulus of complex numbers

Suppose a, b,c three complex numbers as: $|a|=|b|=|c|= 1$. How can I prove that: $\left|\frac{a-b}{1-a\overline{b}}\right| = 1$ and $|ab+bc+ca| = |a+b+c|$
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1answer
32 views

Equivalence between real part of two complex numbers

Suppose $\forall z \in \mathbb{C}\setminus\{i\}$ we set $\displaystyle f(z)=\frac{z+i}{1+iz}$. How can I prove that: $\displaystyle \Re (f(z))=\frac{1}{2} \Longleftrightarrow \Re (z) = ...
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2answers
4k views

Difference between imaginary and complex numbers

Recently I was talking to my teacher about complex and imaginary numbers and he told me basically that $i$ is a complex number; its real part is just 0. However, this has made me wonder; if you can ...
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1answer
476 views

Minimizing l-infinity norm of complex vector

I have an $n$-dimensional complex vector space, and I want to minimize the $L_\infty$ norm of a point that is constrained to an $m$-dimensional affine subspace. That is, Given $\mathbf{z} \in ...
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1answer
72 views

Problems with basic algebra

I'm studying for an exam in a digital communications course I'm taking, and the solution to one question has me totally lost. While finding the Inverse Fourier Transform of a function, there's one ...
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2answers
473 views

Finding a branch of complex logarithm $\log(z)$ with parabola branch cut

Find a branch of $\log(z)$ on domain $\mathbb{C}\setminus T$ where $T=\{x+iy:x\ge 0, y=x^2\}$ I know the branch cut will be a parabola, branch cuts are usually rays, and my prof. did explain it ...
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2answers
308 views

equality of two complex numbers

For $z\in { C }, \Re (z)\neq 2$ we have $F(z)=\frac { 4-z\overline { z } }{ 4-z-\overline { z } }$. I'm trying to prove the equality between the modulus of these numbers without using the ...
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0answers
126 views

Some trigonometric equation problems

show that : $$\left(1+\cos \frac{2\pi}{13}\right)\left(1-\cos \frac{4\pi}{13}\right)\left(1+\cos \frac{6\pi}{13}\right)\left(1+\cos \frac{8\pi}{13}\right)\left(1-\cos ...
1
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1answer
112 views

Prove Using Complex Multiplication

Show the folowing general arctangent identity using complex multiplication, $\arctan\frac{1}{a-b} = \arctan\frac{1}{a} + \arctan\frac{b}{a^2-ab+1}$, for distinct real numbers $a$ and $b$.
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2answers
93 views

Euler's Formula Question.

Find $\cos\theta + \cos3\theta + ... + \cos((2n+1)\theta$, and $\sin\theta + \sin3\theta + ... + \sin((2n+1)\theta$. Where $\theta \in$ Reals.
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3answers
605 views

Complex number inequality.

If z and w are distinct complex numbers such that $|z| =|w| = r$, prove that $|\frac{1}{2}(z + w)| < r$.
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1answer
460 views

Sum of a complex, finite geometric series and its identity

I have the formula for summing a finite geometric series as $$1+z+z^2\cdots +z^n = \frac{1-z^{n+1}}{1-z},$$ where $z\in\mathbb{C}$ and $n=0,1,...$. I am asked to infer the identity $$1+\cos\theta+\cos ...
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3answers
113 views

Placing complex polynomial into Taylor series form

For the following complex polynomial: Write the following polynomials in Taylor form centered at $z=2$: $z^{10}$ How come this simplifies to just a Taylor series of a binomial? Detailed explanation ...
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7answers
470 views

What is the value of $i+i^2+i^3+\cdots+i^{23}$? [duplicate]

Can anyone help me with this question and show me a step by step solution please? The imaginary number is $i$ is defined such that $i^2=-1$. What is $i+i^2+i^3+\cdots+i^{23}$?
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2answers
76 views

Math complex numbers.

I have heard that $i=\sqrt{-1}$ and I have also read about it here http://www.mathsisfun.com/numbers/imaginary-numbers.html. Now I want to ask why in example $\sqrt{-4} = 2i$ as $i=\sqrt{-1}$.
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1answer
198 views

Question on complex number calculation for transmission coefficient of finite potential well

This is actually in my quantum mechanics textbook (pure math question though), and I just cannot see why this equality is true. Any help would be greatly appreciated! Let $F$ and $A$ be nonzero ...
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4answers
219 views

What is the value of $i^i$? [duplicate]

I understand that when you raise any number $x$ to a power, you multiply $x$ by itself the number of times indicated in the power. However, what happens when $i^i$ is performed? How can a number be ...
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1answer
78 views

Sketching complex image

Let f = e^(2-z). Find and sketch the image f(S) of the strip S= { 1 < Rez =< 2, -pi/4 < lmz =< 0} I got radius of f is bound by e^3 =< r =< e^4 , 0 < Argz < pi/4 but the ...
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1answer
68 views

Finding the largest set where a complex function is analytic

f(z) = e^z / (sinz - cosz) So I solved for sinz - cosz = 0 and got pi/4. But why is it Pi/4 + kpi and not Pi/4 + k2pi for the part of the complex plane where this function is not analytic. Thanks.
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0answers
115 views

Complex Logarithm

For what values of $p$ is the following valid? $$\log(z^p) = p\log(z)$$ where $$\log(z) = \ln(|z|) + i[\arg(z)+2\pi n]$$ where $n$ is an integer. I heard the expression above should not be valid for ...
2
votes
2answers
141 views

Complex limits question

Determine the following limit or explain why the limit in question does not exist. $$ \lim_{z \to 1+i} \frac{z^2 - 2z + 2}{\lvert z \rvert^2 - 2} $$ I found this question online and was wondering what ...
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2answers
39 views

Complex sets: factoring into circle

How must $|z|=3|z-1|$ be factored so I end up with a circle, plugging in $z=x+iy$ seems to just up with square roots everywhere. Detailed steps is much appreciated, thanks!
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2answers
421 views

Factor out into ellipse complex numbers

Factor into equation of ellipse the following sets: $$|z+2i|+ |z-2i| =6$$ $$|z-3|-|z+3|=4$$ I got to the part of taking one of the square roots and bringing it to the other side and then squaring ...
2
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2answers
146 views

Prove the relation of $\cosh(\pi/2)$ and $e$

Prove that: $$\cosh\left(\frac{\pi}{2}\right)=\frac{1}{2}e^{-{\pi/2}}(1+e^\pi)$$ What I have tried. $$\cosh\left(\frac{\pi}{2}\right)=\cos\left(i\frac{\pi}{2}\right)$$ ...
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2answers
104 views

Computing with Cauchy Residue theorem

how do I calculate $$\operatorname{Res}\left(\frac{1}{z^2 \cdot \sin(z))}, 0\right)$$ What is the order of the pole? $3$?
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4answers
91 views

rewrite $2ie^{i\pi}+i^3$

i am asked to rewrite $2ie^{i\pi}+i^3$ into $x+iy$ form. i just tried all what i know so far, but couldnot come to solution. i said: $2ie^{i\pi}+i^3=2ie^{i\pi}-i$ but further i am stuck really. i am ...
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1answer
112 views

Complex numbers: negative absolute value (radius)?

I need to find a complex number that represented by the following poler representation: ($\mathbb r$, $\theta$) = ($-5$, $\pi \over 2$) My question is: how is a negative radius (absolute value) ...
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1answer
40 views

Nadirashvili surface (part 2)

The article is 'Hadamard and Calabi Yau conjectures on negatively curved an minimal surfaces' Nadirashvili. In the proof of proposition 4.3 it asserts that the function y is holomorphic. I'm not sure ...
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1answer
149 views

Prove the following equation of complex power series.

Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have $$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$ $$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$ My guess ...
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2answers
440 views

Triangle Inequality with complex numbers: Prove that $ |z_{1} - z_{2}| \le |z_{1}| + |z_{2}| $.

Prove that $ |z_{1} - z_{2}| \le |z_{1}| + |z_{2}| $ for all $ z_{1},z_{2} \in \mathbb{C} $. I don’t quite get how to deal with these triangle-inequality proofs, so any extra insight would be greatly ...
2
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2answers
149 views

Prove by Induction

For $n\in \mathbb{N}$ and $z\in \mathbb{C}$: $\sin{(nz)}=\sum _{ k=0 }^{ n }{ \binom{n}{k} }\frac{1}{2i}(i^k-(-i)^k)(\cos{z})^{n-k}(\sin{z})^k $ $\cos{(nz)}=\sum _{ k=0 }^{ n }{ ...
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3answers
148 views

Cartesian and Polar Coordinate

I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers: a) $z_{1}=-i$ b) $z_{2}=\sqrt{3}+i$ ...
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1answer
291 views

Euler's totient function and complex numbers

I have this problem without a solution (yet) Say I have a complex number and wish to calculate $\phi(a+bi)$ where $\phi(n)$ is Euler's totient function. How would it have to be calculated? I know ...
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5answers
583 views

How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler' formula states that: $e^{i x} = \cos(x) + i \sin(x)$ I can see from the MacLaurin Expansion that this is indeed true, however, I don't intuitively understand how raising $e^{i x}$ power ...
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1answer
343 views

Recursive FFT java implementation

Given below is my java program for FFT. For the input {0,2,3,-1} its returns a false output in complex point representation. ...
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1answer
104 views

Can anyone help me with these questions???

There is questions i am not sure how to tackle... I was wondering if anyone wouldnt mind helping me out with this question? Thanks in advance for any input. first question Find complex numbers $x$ ...
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1answer
129 views

determining if a complex number is a root of unity

How would you determine if $a+ib$ is a $n$th root of unity for some unknown $n$? Obviously the modulus of $a+ib$ must be $1$. But you also need to determine if the $a+ib$ is located at the vertex ...
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3answers
280 views

Where is the mistake in this proof?

I can't figure out, where is the mistake: $$z=re^{i\phi}=re^{\large \frac{2\pi i\phi}{2\pi}}=r(e^{2\pi i})^{\large\frac{\phi}{2\pi}}=r1^{\large\frac{\phi}{2\pi}}=r1=r$$ And we found that the complex ...
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1answer
43 views

Need help with understanding exponential form.

I'm looking at this example in my book: $$z = -1 - i$$ The book doesn't explain how it got to the exponential form, which is: $$\sqrt2e^{-i3\pi/4}$$ I understand how $3\pi/4$ was found, but I don't ...
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1answer
58 views

Linear set of equations with complex numbers

How can I solve a linear set of equations with complex numbers? I haven't solved a set of equation with complex number before, so I'd like to know if there are particular rules to follow.. Thanks for ...
2
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1answer
137 views

A complex square root in the Schrödinger kernel

Consider the initial-value problem for the Schrödinger equation $$\tag{IVP} \begin{cases} i\frac{\partial u}{\partial t}+\Delta u=0 & x\in \mathbb{R}^n,\ t\in \mathbb{R}\setminus \{0\} \\ &\\ ...
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1answer
156 views

Is there a complex variant of Möbius' function?

When you're dealing with arithmetic functions, you might have come across the classical Möbius' function $$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = ...
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1answer
128 views

deriving an identity for complex numbers

For future reference the following question is from Complex Variables and Applications by Brown and Churchill, 8th Edition. Question #6 on Page 8 concerning the derivation of an identity. Let $z = ...
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2answers
196 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
3
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3answers
193 views

The distance from 1 to the other $n$th roots of unity

I want to prove that the sum of the fourth powers of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $6n$. I consider the distance from 1 to the other $n$th roots of unity ...