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

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2answers
73 views

Argument of a complex number

Find the argument $ \theta $ of a complex number z that satisfies the following condition: $|exp (z^3)| \to 0$ as $|z| \to \infty$ I suspect the real part of $z^3$ must be negative and this is enough ...
3
votes
3answers
178 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
2
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1answer
38 views

How is this computed?

$ \left| 1-e^{-is\lambda} \right|^2 = 2 (1-cos\lambda s)$ where $i=\sqrt{-1}$ I don't know how to work with $i$. Thanks
1
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1answer
49 views

Graphing $\sqrt { -x}$

how does my calculator graph ($\sqrt { -x}$. Since I can't graph a complex number, how does my calculator graph the $\sqrt { -x}$ ?
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1answer
112 views

From 'The Joy of x' book: John Hubbard and problems with multiple roots

My math skills are super rusty. In an effort to get some vigor back I started some reading and picked up The Joy of x based on its rave reviews.. I just couldn't make any sense out of the following ...
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7answers
570 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
2
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2answers
112 views

Results Analogous to the Two and Four Square Theorems.

A result that arises out of the study of $\mathbb{Z}[i]$ is that the following are equivalent for integer primes p: 1) $p\equiv 1$ (mod 4) or $p=2$ 2) $\exists a,b\in\mathbb{Z}$ such that ...
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1answer
101 views

Use algebra of Big-O notation to express tan($z$)

We can use the definition of Big-O notation to simply prove that $\sin(z)=z-\frac{z^3}{6}+O(z^5)$ as $z\rightarrow 0$, $\cos(z)=1-\frac{z^2}{2}+O(z^4)$ as $z\rightarrow 0$ and ...
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2answers
59 views

Proving an inequality with the Schwarz inequality

Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let $f$ be a complex value function that is continuous within $0 \le x \le 1$, ...
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2answers
81 views

Complex Numbers - Polar Form/Algebraic Form

I am having some problem with this question: Write the following complex number's in the algebraic form: $\dfrac{5i}{(1-2i)(1-i)(1+3i)}$
2
votes
2answers
88 views

Prove that 1 has n distinct roots of order n

I am trying to show that 1 has n distinct roots of degree n, or in other word that the equations $$z^n=1$$ has n different roots over the complex field. I know that the fundamental theorem of Algebra ...
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1answer
119 views

Dilations - Complex Numbers

Not really sure where to start...hints are appreciated, thanks. In this problem, we will show that the composition of two dilations is, in general, another dilation. (a) Let $ z_0$ be an arbitrary ...
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2answers
311 views

Complex Numbers Geometry

I'm not sure where to begin on this problem - do I plug in for a and solve for z? I was also given a hint: Let z be a point on the line we're trying to describe. We have good tools in complex ...
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2answers
119 views

Solve the equation $x^{2n} + 1 = 0.$ Use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients.

I am asked to solve the equation $x^{2n} + 1 = 0,$ and to use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients. I am given the hint that pairing factors arising from ...
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3answers
39 views

Value of the complex expression

How can I calculate the exact value of something like that: $|e^{\sqrt{i}}|$
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3answers
107 views

Determine all complex number for which: $ \arg(Z^6) = \arg(-Z^2),\ \mathrm{Re}(Z^3) = 2 $

While preparing for the next semester, I stumbled upon this complex number problem which kind of confuses me. I know it has something to do with this - but I simply ...
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1answer
336 views

The strange (for me) case of Mod of Iota.

This might be a silly question to some, but I need some help in this topic. Iota, denoted as 'i' is equal to the principal root of -1. Therefore, $\iota^2 = -1$ When studying Modulus, I was ...
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votes
3answers
361 views

$n$th derivative of $e^x \sin x$

Can someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - ...
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1answer
60 views

Complex Numbers Graphing

The equation of the line joining the complex numbers $-5 + 4i $ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38 $ for some complex numbers a and b. Find the product ab. Well if ...
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2answers
42 views

Is convolution associative with regards to the complex unity?

Setup: I need to do a convolution with the function $\cfrac{i}{x}$, and I would like to get rid of the $i$. My functions to be convolved are all real valued. According to the ever-failable wikipedia, ...
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0answers
20 views

Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$

This is a question from a book that I'm trying to solve and I don't know how. Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$ Can you please give me some ...
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1answer
76 views

A complex question on complex numbers

Let$$w=\frac{\sqrt3+i}{2}$$ and $P=\lbrace w^n:1,2,3,...\rbrace$. Further $$H_1=[{z \in}\mathbb C:\text{Re}\,z\gt \frac{1}{2}]$$ and $$H_2=[{z \in}\mathbb C:\text{Re}\,z\lt \frac{-1}{2}]$$,where ...
3
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4answers
1k views

Weird stuff happening with complex numbers on a ti-84

So, I'm trying to do some calculations for my Electrical engineering homework. This requires a bit of algebra with complex numbers. I have been finding that some of the calculations that my calculator ...
2
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1answer
92 views

What properties do we lose when moving from the rational numbers to the real numbers?

When we pass from the real numbers to the complex numbers, we lose total ordering. But what do we lose when we move from the rational numbers to the real numbers?
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1answer
29 views

Linear Algebra Complex values

The cube roots of $-3+2i$ are $x_1 = (1.0106+1.1532i),\; x_2 = (0.4934-1.4519i),\text{ and }x_3 = (-1.5040+bi)$ What is $b$? So $$-3+2i = (x_1)(x_2)(x_3) = -3.268 + 2.172bi + 1.351i + 0.898b$$ ...
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3answers
315 views

Linear Algebra Complex Numbers

The solutions to the equation $z^2-2z+2=0$ are $(a+i)$ and $(b-i)$ where $a$ and $b$ are integers. What is $a+b$? I simplified and got $(z+1)(z+1) = -1$ and now I'm not sure where to go from there. ...
2
votes
2answers
76 views

Evaluate $a^i$ where $a$ is real:

I have a question involving the evaluation of $3^i$, but I am unsure how to do this. I know how to solve such questions involving $e^{i\theta}$, but how does this work with a different base? (I ...
2
votes
3answers
63 views

Residue of $\frac {z}{z^6+1}$ at $z=i$

Is there an easy way to find the residue of $$\frac {z}{z^6+1}$$ at $z=i$? the formula given by taylor theorem doesn't seem to help and evaluating $(z-i)\frac {z}{z^6+1}$ at $z=i$ seems to laborious. ...
3
votes
2answers
99 views

An apparently elementary fact about complex numbers.

Let $f,g\in \mathbb{C}$. My textbook states that it is an elementary fact that $$\lim_{t\to 0,t\in\mathbb{R}}\frac{|f+tg|^p-|f|^p}{t}=\frac{p}{2}|f|^{p-2}(\overline{f}g+f\overline{g})$$ I don't know ...
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3answers
95 views

how to solve circle dividing equation (complex numbers)

I have a equation that should divide a circle in even parts. As I found its called circle-dividing equation. I'v found same information how to solve a equation which has a form like this: $$z^6 = ...
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2answers
579 views

Application of maximum modulus principle

Let $f$ be a non-constant holomorphic(analytic) function in the unit disc$\{|z|<1\}$ such that $f(0)=1$ then it is necessary that $(1)$ there are infinitely many points $z$ in the unit disc such ...
0
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2answers
142 views

Fourth roots of $-17$

The question I'm stuck on is as follows: Find all $4$th roots of $−17$ in Cartesian form. Simplify as much as possible. Here's what I've done so far: $$z^4 = -17\\ |z| = \sqrt{(-17)^2} = 17 = ...
2
votes
3answers
58 views

When are complex conjugates of solutions not also solutions?

I've heard that for "normal" equations (e.g. $3x^2-2x=0$), if $(a+bi)$ is a solution then $(a-bi)$ will be a solution as well. This is because, if we define $i$ in terms of $i^2=-1$ then we might as ...
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2answers
34 views

Trouble on proving a lemma - Complex Power Series

I'm having some difficulties on proving the following lemma: "If $f_n$ is a sequence of functions which converges uniformly to $0$ on a set $G$ and $z_n$ is any sequence in $G$ then the sequence ...
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1answer
54 views

Integration of a complex number

I'm trying to integrate, for example, $\int e^xe^{-inx} \,dx$. $i$ is the imaginary unit, $n$ is a constant. I tried to integrate normally - as I would in a Real number: $$\int e^x e^{-inx}\,dx = ...
2
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0answers
38 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
2
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1answer
72 views

$A+A^T=I$, $\lambda$ is an eigenvalue of $A$, show that $\lambda=\frac{1}{2}+\alpha i$

I tried to solve it but I got $\lambda =\frac{1}{2}$ without the complex part, I'd like to know where my logic is flawed. Assume $v$ is the eigenvector associated with lambda, then: $(A+A^T)v=Iv$ ...
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0answers
93 views

Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
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1answer
56 views

Find the residues of $f(z) = \left( \frac{z-1}{z+1}\right)^{\frac{1}{2}}\frac{1+z} {1+z^{2}}$

Consider the function $$f(z) = \left( \frac{z-1}{z+1}\right)^{\!\frac{1}{2}}\frac{1+z} {1+z^{2}}$$ I want to calculate the residues of $f$ in $\{+i,-i\}$. Using the usual techniques, we have that ...
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2answers
140 views

Roots of Unity - Complex Numbers

The sets $A = \{z : z^{18} = 1\} $and $ B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. ...
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2answers
95 views

How to compute the norm of a complex number under square root?

How to compute the norm of a complex number under square root? Does the square of norm equal the norm of square: $\|\sqrt z\|^2 = \|\sqrt {z^2}\|$? Let $z = re^{i\theta}$, then $$\|\sqrt z\|^2 ...
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2answers
30 views

complex number congjugate criteria

Is it true that if a complex number $z_2$ times $z_1$ is the square of norm of $z_1$, then $z_2$ is the conjugate of $z_1$? $z_2 = \bar{z_1} \Leftrightarrow z_1z_2 = \|z_1\|^2?$ It occurs to me to ...
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1answer
90 views

Elliptic curves: Why over the complex numbers?

I am a math undergrad, so much of the literature on elliptic curves escapes me. I'm trying to understand why one considers elliptic curves over the complex numbers. Specifically, this part of the ...
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1answer
27 views

Proof that complex conjugate of polynomial result equals pynomial result with complex conjugated argument

This question feels uneasy to be expressed by words for me, however, I'm asked to prove this: $$P(\overline{a+bi}) = \overline{P(a+bi)}$$ Of course, $\overline{a+bi} = a-bi$.
4
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3answers
287 views

Solve $\cos(z)=\frac{3}{4}+\frac{i}{4}$

I tried solving this using the definition of $cos(z)=\frac{e^{iz}+e^{-iz}}{2}$ and equating it to $\frac{3}{4}+\frac{i}{4}$ and converting it to a complex quadratic equation through a substitution ...
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1answer
584 views

equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
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2answers
41 views

finding a solution for $m$ given $(1+i)z^2-2mz+m-2=0$

Given the equation: $(1+i)z^2-2mz+m-2=0$, while $z$ is complex and $m$ is a parameter. For which values of $m$ the equation has one solution? So my idea was to use: $b^2-4ac=0$ for $ax^2+bx+c=0$ ...
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5answers
2k views

Finding real and imaginary part of exponential function

Can someone explain to me how I find the real and the imaginary part of $e^{\theta i}$? I'm learning complex numbers but I don't quite understand how $e$ is intertwined in all this.
1
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1answer
70 views

Complex root won't work

So I'm trying to get this: http://www.wolframalpha.com/input/?i=%288*sqrt%283%29%29%2F%28z%5E4%2B8%29%3Di And I've calculated $z^4=16 \left( \cos (\frac{- \pi}{3})+ \sin ( \frac{- \pi}{3}) \right)$ ...
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2answers
41 views

Complex roots problem [duplicate]

I've got a complex equation with 4 roots that I am solving. In my calculations it seems like I am going through hell and back to find these roots (and I'm not even sure I am doing it right) but if I ...