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

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What are the properties of the fourier transform of a phase-only function?

Given a function of the form: $$ f(x) = e^{i\phi(x)} | \phi(x)\in\Re $$ What are the properties of its Fourier transform? For instance, purely real functions have Fourier transforms with symmetric ...
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25 views

For wich $u\in\mathbb{C}$ is $0^{u}$ defined?

It is obvious that $\left|e^{v}\right|=e^{\text{Re }v}>0$ showing that $\ln z$ is not defined for $z=0$ . So the expression $z^{u}=e^{u\ln z}$ cannot be used here. Nevertheless we don't hesitate ...
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146 views

Correct way to write the polar form of a complex number

What is the most correct way to write the polar form of a complex number? For example, given the complex number: $\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i$ I would write the polar form as follows: ...
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1answer
36 views

How find this inequality find the maximum $z_{5}$

let $z_{1},z_{2},z_{3},z_{4},z_{5}\in C$,such $$\begin{cases} |z_{1}|\le 1,|z_{2}|\le 1\\ |2z_{3}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ |2z_{4}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ ...
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1answer
66 views

Complex Numbers-Locus

prove- $|z-z_1|^2 +|z-z_2|^2=k $will represent a circle if $|z_1-z_2|^2\leqslant 2k$. $z,z_1,z_2$ are complex numbers. What i have tried out- $|z-1|^2+|z-2|^2=k$ let $z=x+iy$ $$\\ ...
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28 views

argument of a lacunary sum of roots of unity

Let $q>4$ and $t< \sqrt{q}$ be integers. Determine the set $\{j_1,...,j_t\}$ of integers $0 \leq j_i <q-1$ such that $\arg(\sum_{i=0}^t e^{2i\pi\frac{j_i}{q}} ) \in [0,\pi[$.
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162 views

sum of $p$th roots of unity

Let $p$ be an odd prime. Let $\mathbb Z_p$ be the ring of integers from $0$ to $p-1$, i.e. $\mathbb Z_p=\left\{0,1,\dots,p-1\right\}$. Let $r$ be a positive integer. Then, for all values of $p$, is ...
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152 views

Complex Limits HW Help

Prove the following limits by using $\epsilon$ and δ 1) Show $\lim_{z\to 2}z2 + iz = 4 + i2$. 2) Show $\lim_{z\to -i} 1/z = i$. 3) Show $\lim_{z\to4i}z/\overline{z}=-1$. For 1) i'm stuck at ...
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216 views

Proving Question (Complex Numbers, De Moivre's)

Prove that $(1+\cos \theta + i \sin \theta)^n + (1+\cos \theta - i \sin \theta)^n=2^{n+1}\cos^n\frac{\theta}{2}\cos\frac{n\theta}{2}$ I want to avoid using the $e^{i\theta}$ form since I haven't ...
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59 views

Regarding Logarithmic of complex numbers

$\newcommand{\Log}{\operatorname{Log}}$ In my undergraduate complex analysis textbook, it claims that $$\Log(1+i)^2=2\Log(1+i)$$ I am not sure if this is a misprint or there is actually a way of ...
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124 views

Simplifying a product over roots of unity

Let $\zeta_{n}=e^{2\pi i /n}$ be the nth root of unity. Now consider the product : $$\prod_{k=1}^{n-1} (1-\zeta_{n}^{k})^{\zeta_{n}^{k}}$$ Is there a simple formula for this product as a function of ...
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130 views

Möbius tranformation

We define a Möbius transformation through: $$z\rightarrow \frac{az+b}{cz+d}, ad-bc\neq0, a,b,c,d\in \mathbb{C}$$ and extend to the Riemann sphere as follows: if $c=0$, $T(\infty)=\infty$, and if ...
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222 views

Möbius tranformation are continuous?

We define a Möbius transformation through: $$z\rightarrow \frac{az+b}{cz+d}, ad-bc\neq0, a,b,c,d\in \mathbb{C}$$ and extend to the Riemann sphere as follows: if $c=0$, $T(\infty)=\infty$, and if ...
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50 views

Polynomial $p(z)$ in $\mathbb{C}$ has no zero whose modulus does not exceed $1$ [duplicate]

For $n>1$ consider real numbers $c_0>c_1>.....>c_n>0$. Prove that the polynomial $$p(z) = c_0+c_1z+.....+c_nz^n$$ in $\mathbb{C}$ has no zero whose modulus does not exceed $1$.
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418 views

Maximum of an absolute value complex function

I'm working my way through Marsden's Basic Complex Analysis book and I can't solve this problem. It's problem 23 of section 1.2 if that helps. Let $a$ be a complex number, find the maximum of ...
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1answer
71 views

how I can prove this summation?

prove that $$\left| \sum_{k=1}^{n} a_k z_k \right|^2 = \left(\sum_{k=1}^{n} \left| a_k \right|^2 \right)\;\left(\sum_{k=1}^{n}\left| z_k \right|^2 \right)-\sum_{1\leq j< k\leq n}\left| ...
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1answer
153 views

How can the lyapunov exponents for the Mandelbrot Set be computed?

I am trying to find a way to calculate the Lyapunov exponents of the Mandelbrot set. There are some very nice diagrams that you can find on Flickr of a plot of the Lyapunov exponents of the Mandelbrot ...
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52 views

Instead of iteration method, can we solve this matrix computation?

I have a simple problem with matrix compuation. Matices what I have are $A_f$ matrix with $(M \times N)$, $B_f$ matrix(or vector) with $(N \times 1)$. I just want to calculate $|A_fB_f|^2$ for $F$ ...
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63 views

Finding $i^{i^i}$. [duplicate]

Express $i^{i^i}$ in the form $a+bi$ where $a,b$ are real. From euler's formula, I get $\ln i=i\pi/2$, which leads to $i^i=e^{-\pi/2}$. Therefore, $\ln i^{i^i}=i^i\ln i=i\pi ...
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111 views

Solving partial fraction involving complex numbers

This is the image of a part of the problem that I am doing. I understand partial fractions fairly well. To solve for A, you will zero out the other term(s). So, For A, if I let B = 0, I am left with ...
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1answer
147 views

Analog to bisection: Converging on complex roots of a polynomial

I am working on a Perl module that, among other features, will solve all the zeroes of a polynomial. Thus far, I am doing OK for $2$, $3$, $4$th degree using quadratic, Cardano's and Ferarri's ...
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67 views

How do I express a complex number in a complex base?

I came across an old mathematical paper on the web, published in the 1980s (I can't seem to find it again). The paper was about complex number arithmetic, and it talked about expressing complex ...
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93 views

Exponential functions of a complex number. I can't get it right.

Question: Prove that $\frac{e^Z}{e^W} = e^{z-w}$ My Attempt: The given equation on left-hand side can be re-written as $e^z.e^{-w}$. Let $z = x + iy$ and $w = u + iv$. $exp(w) = e^u(cosv+isinv)$. ...
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2answers
106 views

Factoring any single-variable polynomial in $\mathbb C$

The fundamental theorem of algebra says $$ \forall p(x):\mathbb C \to \mathbb C,\ p(x) = a\prod_{n=0}^m\big(b_nx+c_n\big) $$ where $p(x)$ is a single-variable polynomial, and $\{a;m\}\cup\{\forall ...
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1answer
41 views

BesselI and BesselK with complex arguments

I would like to check with you the following issues. During my work, I ended up with following Bessel Functions. BesselI(0, i*x) and BesselK(0, i*x). (Modified bessel functions of first and second ...
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19 views

simplification of a complex expression

I am collecting proofs for certain integrals. To simplify certain proofs, I use $e^{Ax}cos(Bx)=\mathcal{Re}[e^{(A+iB)x}]$, where $A$, $B$, and $x$ are real. Is there an analogous simplification for $ ...
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38 views

nasty exponentials

While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like $$\large \rm ...
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97 views

Rotation of angle $\frac{\pi}{4}$ about the point $i$

Need to find an isometry which would rotate about the point $i$ by $\frac{\pi}{4}$. So I was thinking that first I return the given point to orign, make the rotation and then translate back, right? ...
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1answer
64 views

Contour Integrals for positively circular contour

Find the contour integral of $\frac{1}{(z^2+1)^2}$ for the positively oriented circular contour $|z-Ri|=R$, for every positive real number $R>\frac{1}{2}$. I don't know how to set up the ...
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34 views

Compare one real number to one complex number. [duplicate]

I understand that complex numbers can be neither ordered nor compared by 'size', but if mapped one for one by a transformation, then they can be. Latter point aside, can I say that $2-xi < 2 < ...
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70 views

Inequality with complex numbers and a power

I am working on a problem in nonlinear analysis, and I would like to estimate a term that I can write, in abstract form, as follows: $$ \left||z|^{2p-2}(\Re (z \overline{h}))^2 - |w|^{2p-2} (\Re (w ...
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70 views

About growth rate of the iterated exponential on the complex plane.

Let $n$ be a positive integer. Let $f(n,x) = exp(f(n-1,x))$ and $f(0,x)=x$. Let $Q(f(n,x)) =1$ if $Re(f(n,x))<2$. Let $S(n,x) = \Sigma_{1}^{n} Q(f(n,x))$. How to estimate $S(n,2+i)$ efficiently ? ...
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127 views

Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
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53 views

Complexity of products/quotients of sums of roots of unity

This question was inspired by another recent question (here) . That older question asked somehow vaguely if the expression $\prod_{k=1}^m \tan(\frac{k\pi}{4m})$ can be simplified (for $m=45$). I ...
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132 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 ...
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121 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 ...
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101 views

Hadamard regularization isn't working out

As part of an exercise in a grad course on "mathematical methods" (always such a helpful name), I've been asked to evaluate $I=\int_0^{1/2}{(x^2-x+c)^{-2}dx}$ as a Hadamard finite part integral for $0 ...
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112 views

Inequalities with $\sin(z)$, $z \in \mathbb{C}$

An exercise asks to find all $z\in\mathbb C$ such that $|{\sin z}|\leq 1$ and then an $n\in\mathbb N$ such that $|\sin(in)|>10000$. Here are some results we can use. For all $z=x+iy\in\mathbb C$ ...
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52 views

show this summation hold in term of integral

i can show $\sum |x|^2=\int_a^b|f(x)|^2dx$ in term of integral, or this one $|\sum x\overline y|^2=|\int_a^bf(x)\overline {g(x)}dx|^2$ but i don't know how to show this one ...
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112 views

Complex numbers - finding a square root of something

Let $z_1 , z_2 $ be two complex numbers that satisfy: $\dfrac{z_2 } {\bar{z_1}}= \frac{3}{8} \big(\cos(75^{\circ})+i\sin(75^{\circ})\big) $ , $z_1 z_2 ^2 = \frac{1}{3} \big(\cos(120^{\circ}) + ...
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1answer
96 views

Complex numbers- true of false?

If f(z) is an entire function, which gets only real values for real z, and $$ f(0)=0,$$ $$f'(0)\ne 0$$ and the Image of the imagainary axie is a straight line, then this line is the imagainary axie. ...
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406 views

The general recipe for finding the conjugate of a complex function

I have the general recipe for finding the complex conjugate of a function down as follows: Suppose I have $f(z)$: Separate $f(z)$ into a sum of real and imaginary functions such that ...
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128 views

Complex Numbers geometry question

on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals). I need to use complex numbers to prove ...
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169 views

Question involving some roots of the equation $31z^{15} - z^{10} + 32 = 0 $

Consider the equation $$31z^{15} - z^{10} + 32 = 0.$$ What would be the sum of all those roots of the equation whose real part is positive? Only trivially trying to solve the equation I find not ...
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129 views

What should be the range of the principal value of a complex number?

My lecturer stated that the principal argument is $[0,2\pi)$. However my tutor (my tutor is different to my lecturer) states that it is $(-\pi, \pi]$. Wikipedia article on complex numbers mentions ...
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55 views

Proving that two representations of a Fourier series are the same

I have to show that $$\sum_{n=0}^\infty A_n\cos\left({xn\frac{2\pi}{T}-\theta_n}\right) \equiv \sum_{n=-\infty}^\infty c_n \mathrm{e}^{\left({ixn\frac{2\pi}{T}}\right)}$$ I have tried two ...
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188 views

Cauchy's formula: $\int_\gamma \frac{e^{z}}{z(z-3)} dz $

trying to compute the integral $\int_\gamma \frac{e^{z}}{z(z-3)} dz $, where $\gamma:[0,2\pi]\to\mathbb{C}, \gamma(\theta)=2e^{i\theta} $ but not sure where to begin. I know, from Cauchy's formula, ...
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4answers
116 views

Need help to simplify the expression involving powers

$$\left(1-\frac{\sqrt{3}-i}{2}\right)^{24}$$ somehow this should be equal to :$$\left(2-\sqrt{3}\right)^{12}$$ but I can't see how...
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36 views

I need to describe a behavior of a curve depending one one parameter in z-plane

$|z^{2}-1|=\lambda$ My aprouch: taking $z=x+iy$ $$z^{2}=(x^{2}-y^{2})+i(2xy)$$ Then : $$|z^{2}-1|^{2}=x^{4}-2x^{2}y^{2}+2y^{2}-2x^{2}+y^{4}+1+4x^{2}y^{2}=\lambda^{2}$$ Now taking ...
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152 views

Real, imaginary parts of $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$

For the function $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$ , show that; $\qquad\mathrm{Re}\,G(w)=\sqrt2\sin^2(w/2)\quad$ and $\quad\operatorname{Im}\,G(w)=-1/\sqrt2\sin w$. I really need help with ...