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

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How to separate out real and imaginary terms

I have an equation like this: $$a+ib = \log(x+iy).$$ I need to separate the real and imaginary part in RHS so that I can equate the real part of LHS to real part of RHS and imaginary to imaginary ...
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1answer
38 views

Find all complex solutions to the equation

i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0 I basically have no clue, any tips/advice/solutions would be great. I could also need some help with another question, this one ...
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2answers
114 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
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2answers
46 views

quotient of complex numbers?

so I was wondering if you have two different equations having denominators $2+i$ and $2-i$ respectively how came the denominator of the quotient in standard form is $5$ for both equations? I tought ...
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1answer
22 views

Prove that the functions $g_k(z) = f_k \circ h_k(z)$ form a normal family.

I am having a bit of trouble with the following complex analysis question which originates from a qual. Some help would be awesome. Let $f_k :\mathbb{D} \rightarrow \mathbb{C}$ be a normal family of ...
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1answer
19 views

Argand Diagram confusion

While searching on net about Argand diagram or complex plane we get images of both kinds which have their y-axis either real axis or imaginary axis. Given z = x + yi, some people write Im(z) = y and ...
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0answers
34 views

Basic complex geometry: Reflexion by a line. Where does $ \overline{z}$ go?

Where does $ \overline{z}$ go in the end? Shouldn't the formula be $f(z)= w^2 \overline{z} +2isw$? Thanks in advance. Original link: ...
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3answers
61 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
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1answer
69 views

Can there be a complex line?

In an early math class, I was shown how all Reals could be constructed from Rationals using a 2-D representation (ex. Real numbers are represented by (a + b \sqrt{2} ) where a & b are Rational). ...
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3answers
32 views

Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
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1answer
109 views

Show $f$ is analytic if $f^8$ is analytic

This is from Gamelin's book on Complex Analysis. Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the ...
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3answers
70 views

Can some one explain to me what is going on here - power of complex number

So here is the question and the work to solve it, but I have no idea how one knows to do the first step or what the first step is... $$ \begin{align} (6-i\sqrt{12})^{12} &= ...
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3answers
37 views

Finding three roots of a complex number if we already know one root

If we know that $a+bi$ is one of the forth roots of the complex number $z$, how can we find the other three roots?
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3answers
33 views

need help explaing the complex roots of a cubic

I am trying to understand a Galois theory example and we are looking at the solutions of $x^3-2=0$. It says they are $2^\frac{1}{3},2^\frac{1}{3}\omega, \text{ and } 2^\frac{1}{3}\omega^2$. I know ...
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2answers
25 views

How to solve these three complex-numbers equations

I'm so stuck right now studying for my bachelor of science math exam. Please show (using i = imaginary unit): 1) i^(1/i) = e^(PI/2+k*2*PI) 2) (4*i)^(1/2) = { 2^1/2*(1+i) ; 2^1/2*(-1-i) } 3) i^(i*pi) ...
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1answer
39 views

Proving that $f(z)$ is polynomial [duplicate]

Given $R>0 , M>0$ Let $f(z)$ be a entire function such as $|f(z)|\leq M|z|^m$ for evey z such as $|z|>R$. Show that $f(z)$ is a polynomial of degree m or lower.
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1answer
47 views

Prooving that a complex function is constant

Let $f(z)=u(z)+v(z)i$, $u(z) \leq 0 , \forall z \in \mathbb{C}$, and $f(z)$ is entire. Then $f(z)$ is constant. The hint is "use the Liouville theorem". I tried , but i need prove which f is limited ...
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1answer
26 views

What is the best way to express a complex modulus squared in text?

I am looking for a way to describe the fact that I am taking a modulus squared in text. E.g. "inserting the potential (Eq. 1) into the expression for the wavefunction coefficients (Eq. 2) and taking ...
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1answer
19 views

Complex function and Jacobian matrix

Given some complex-differentiable function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined $f(x,y)=u(x,y)+iv(x,y)$, we know the Cauchy-Riemann equations hold, so: $$\dfrac{\partial u}{\partial ...
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1answer
19 views

About the convergence or divergence?

Whether the following integral converge or diverge by comparison test. \begin{align*} ...
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3answers
47 views

How can one prove $||z_1|-|z_2||\le|z_1+z_2|$?

How can it be shown that for the complex numbers $z_1$ and $z_2$: $$||z_1|-|z_2||\le|z_1+z_2|$$ My text provides a hint that $z_1=z_1+z_2+(-z_2)$, and $z_2=z_1+(-z_1)+z_2$. $${}$$
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2answers
60 views

Expansion of $\sin^5 \theta$ using the Complex Exponential

How do I expand $\sin^5\theta$ using the complex exponential, in order to obtain: $$\frac{1}{16}\sin 5\theta - \frac{5}{16}\sin 3\theta + \frac{5}{8}\sin\theta$$ Thank you.
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4answers
65 views

$ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and no idea about $\sum\limits_{n=1}^\infty \dfrac{(3+4i)^n}{5^n\,\sqrt[999]{n}}$

How can one see that $ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and by which criterion? I was using a binomial theorem for $ (1+i)^n $ as $ \sum\limits_{k=0}^n \dbinom{n}{k} i^n$, ...
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6answers
216 views

Is $i$ irrational?

On the one hand, $i(=\sqrt{-1})$ cannot be expressed as a ratio of integers, so, by definition, $i$ is not rational $\iff i$ is irrational. However, the set of irrational numbers, ...
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2answers
48 views

How to solve this equation? $8z^3+1=0.$ [closed]

I would like to know how to do this question. (a) Determine all of the solutions of the equation $$8z^3+1=0.$$ Express your solutions in the form $z = re^{i\theta}$ where $-\pi<\theta \leq \pi$. ...
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1answer
19 views

Draw a set of values in complex plane where the complex number $w=1-3i$ is pure imaginary number.

How would you draw a set of values (in complex plane) where the complex number $w=1-3i$ is pure imaginary number? Could this be the solution? If $Rew=0$.
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1answer
43 views

How to solve the complex equation? $(x+2yi)^2 = xi.$

How to solve the following complex equation with in less than 60 seconds? $$(x+2yi)^2 = xi.$$ I know how to solve, we have to solve power first then real part equal to real part and imaginary to ...
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0answers
34 views

Schwarz's Lemma and extensions in complex analysis

I was assigned this problem: (which I here present verbatim) Let $f$ be a holomorphic function of the unit disk unto itself. Prove that $|f'(0)|\le 1$. Isn't it also necessary to assume that ...
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2answers
83 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
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2answers
43 views

Prove that $z_1, z_2, z_3, z_4$ are the vertices of a rectangle if and only if…

I have to prove that $z_1, z_2, z_3, z_4$, where $|z_1| = |z_2| = |z_3| = |z_4| = 1$, are the vertices of a rectangle if and only if $z_1z_2z_3+z_1z_2z_4+z_1z_3z_4+z_2z_3z_4=0$ Any help? There is a ...
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1answer
50 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
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1answer
36 views

Complex summation simplification

What I'm getting is $$\frac{( \sin (N+1)x - 2^N \sin x)}{(2^N(\sin x - 2))}$$ How do I simplify to the form they have given , please help. I hope it's clear because I don't know Ajax still ...
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1answer
30 views

Summation of complex numbers and simplification

By considering $$ \sum_{k=0}^{n-1}(1+i\tanθ)^k\tag{1}$$ Show that $$ \sum_{k=0}^{n-1}\cos(kθ)\sec^kθ=\cotθ\sin(nθ)\sec^nθ\tag{2}$$ Provided $θ$ is not an integer multiple of $\frac{π}{2}$. My take ...
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3answers
52 views

System of equations with complex numbers

This might seem quite trivial for people who are knowledgeable in complex analysis, but it is not so much to me. I am trying to find an efficient way to solve the following system of equations: $$ ...
2
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1answer
25 views

absolute value of sum of complex numbers squared

is this correct $ \left| |a| \exp(-i c)-|b| \exp(-i d) \right|^2=|a|^2-2|a||b|+|b|^2$ Thank you
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1answer
48 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
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1answer
51 views

Is this function familiar to anyone?

Consider $$f(z)=\sum_{w\in C}\frac{1}{z-w}$$ Where $C$ is the set of complex integers. What I would like to know is where can I find any information about this function (name perhaps). For instance, ...
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1answer
34 views

Solution to polynomial over complex numbers [closed]

Solve $$z^6 = 8z^3-7z$$ over the complex numbers by considering the roots of complex numbers.
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4answers
153 views

$e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?

My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. It seems to me that the formula is a ...
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2answers
55 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
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1answer
37 views

Find the real and imaginary parts of $\sin(\frac{\pi}{2}+i\ln2)$

Find the real and imaginary parts of $$\sin\left(\frac{\pi}{2}+i\ln2\right)$$ Using the double angle formula I have gotten ...
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1answer
49 views

Solve $(\frac{z+1}{z})^5 =1$ using fifth roots of unity

$$(\frac{z+1}{z})^5=1$$ Show that its roots are $$-\frac{1}{2}(1+i\cot(\frac{kπ}{5})), k = 1,2,3,4$$ I need to use the five fifth roots of unit, with angles $0,\frac{π}{5}, ...
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8answers
711 views

Is $e^{i\pi}+1=0$ all it's cracked up to be?

While it is beautiful and elegant and all that, isn't it true that Euler's identity is really just an artifact of how we define the radian? I'm speaking of those who say that it's great because it ...
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2answers
54 views

Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$

I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you. Let $f$ be a polynomial such that $|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$ for ...
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2answers
77 views

What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
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1answer
88 views

Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$

This is an old qualifying exam problem that I am working on. I would appreciate some help. Thank you. Prove that there exists an analytic function $f : D → D$ such that $f(1/2) = f(−1/2)$ and ...
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1answer
40 views

Proof using de Moivre's Theorem

Let $z=\cos\theta + i\sin\theta$ Show that $$1+z = 2\cos\frac{\theta}{2}(\cos\frac{\theta}{2}+i\sin\frac{\theta}{2})$$ I don't even know how to start on this proof.
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1answer
23 views

An inequality on the real part of a square root

I have the following inequality: $\Re(k+z) \geq \Re \sqrt{(k+z)^2-4z}$ where $k$ is real and $z$ complex. Under what conditions on $k$ and $z$ is this inequality true? I suspect that it is true for ...
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1answer
59 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
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1answer
43 views

De Moivre and trignometry question

I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 ...