Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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1
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2answers
57 views

Why cannot the power of $i$ be negative?

In a question, The penultimate step of the result was $(i)^n = 1$, and it required to figure least value of n. I checked the $-4$ option, but it said the answer was $4$. Why so? The complete question ...
-2
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0answers
48 views

Finding complex eigenvectors

Can anyone help me to point out what I am doing wrong? I need to find a change of bases matrix for the complex eigenvalue (so I can find closed formula). I was successful in finding eigenvalues, ...
0
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2answers
20 views

Simplyfing complex expressions with square roots

Simplify the expression $z =(4+4 \sqrt 3 i)^{1/2}$ so that it's in the form $z = x + iy$. So far I got: $$4(1+ \sqrt3 i)^{1/2}$$ But I'm unsure where to go next. I don't know you how can ...
0
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1answer
23 views

$\left | \sum_{n\in \mathbb N} a_n b_n z^{n} \right | \leq C \left | \sum_{n\in \mathbb Z} b_n z^n \right | (z\in \mathbb C)$?

Let $ a_n , b_n \in \mathbb C$ for all $n\in \mathbb N.$ And there is $M>0$ such that $|a_n| \leq M$ for all $n\in \mathbb N.$ Can we expect $\left | \sum_{n\in \mathbb N} a_n b_n z^{n} \right | ...
2
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1answer
48 views

What we can tell about complex matrices? Ideas for a school work

Background: I have some background in abstract and linear algebra. In my undergrad complex calculus class, I have to write a $5$ page paper about "complex matrices". I don't know exactly what the ...
0
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1answer
31 views

Find the modulus and the principal argument of $-2i$

Let, $z=0+(-2i)$ $\therefore$ mod of $z=2$ But, I am getting stuck over here and I am unable to find the argument as the $\tan\alpha$ comes out to be not defined. Any hint or help would be much ...
0
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1answer
20 views

Showing complex transformations in a fluid way

It says it all in the title: I need to show how simple complex transformations (translations and dilations, or even both) affect shapes on the complex plane in a "fluid" way – that is, creating some ...
4
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0answers
41 views

Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
1
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2answers
21 views

Max/Min modulus principle.

Let $f$ be analytic function inside and on a bounded domain $D$. If $\Re(f)$ is constant on the boundary, then $f$ is constant in $D$. I realized that maximum value of $\Re(f)$ must occur on the ...
0
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0answers
16 views

Affine transformations in the complex

In $\mathbb C^2$ I have the following three lines: $r_1:3x-y+3=0, r_2:y=0, r_3:x-i=0$ I want to find all the affine transformations such that $f(r_1)=r_2, f(r_2)=r_3, f(r_3)=r_1$ How can I do it? ...
0
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2answers
42 views

nth roots of the polynomial $x^3 =2$

I have to find the solution of the polynomial x^3 - 2 =0. Attempt: $x^3=2$ $x^3=2.1 =2(cos2k\pi+isin2k\pi)$ and $k=0,1,2$ $x=2^{1/3}(cos(\frac{2k\pi}{3})+i sin(\frac{2k\pi}{3}))$ now we will get ...
4
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3answers
62 views

Complex polynomial P with $P(n)= (-1)^n$

I want to show that there is no polynomial P with complex coefficients such that $P (n) = (−1)^n$ for all integers n.Does there exist an entire function with this property ? Thank you.
0
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2answers
38 views

Maximum value of $\frac{\alpha\overline{\beta}+\overline{\alpha}\beta}{|\alpha\beta|}$

Maximum value of $$\frac{\alpha\overline\beta+\overline\alpha\beta}{|\alpha\beta|}$$ is 1) 2 2) 1 3) none of the above. Considering $\alpha=x+iy$ and $\beta=m+in$ , on evaluating the expression I ...
3
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1answer
68 views

Solutions of $ \tan(z) = \frac{z}{z^{2} + 1}$ in the complexes

In an exam I got this question: Show that if the equation $$ \tan(z) = \frac{z}{z^{2} + 1} $$ has $z_{0}$ as a solution, then $ \Re(z) = 0 $ or $ \Im(z) = 0 $ Writing $z$ as $x + i y$ seems too ...
0
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1answer
21 views

When is the radius of convergence of the product of two complex power series twice the radius of convergence of the product of the radii

Proving that the product has a larger radius then the product isn't too bad using the nth root test, but another practice question I have asks for examples of power series $\sum a_kz^k$ with radius of ...
0
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0answers
20 views

randomly generate M pairs of complex numbers from 1 to N, find gcd.

I want to write a script to generate statistics on gcd's and number of steps required to find them by the Euclidean algorithm, using M randomly generated pairs $a+bi$ between 1 and N. And plot them. ...
2
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0answers
46 views

Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly

I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ ...
7
votes
2answers
106 views

Find $\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$

Find $$\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$$ The general term is $\frac{1}{r}\sin\frac{r\pi}{3}$ Let $z=e^{i\frac{\pi}{3}}$ Then, ...
2
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1answer
54 views

Finding residues at a point $a$ where $a$ is a pole.

I am faced with the following problem: (a) Find $\displaystyle res_{a} \frac{\varphi(z)}{(z-a)^{n}}$ where $\varphi$ is a given function analytic at $a$, $\varphi(a) \neq 0$, and $n$ is a positive ...
0
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1answer
51 views

Can I express some power of $\cos(\frac {2\pi}{5})$ as a rational number without using complex numbers?

I have been trying to express a power of $\cos(\frac {2\pi}{5})$ as a "rational number", or trying to find a "rational number" that results from a linear combination of powers of $\cos(\frac ...
0
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1answer
35 views

Residue of $1/(\sin(1/z))$ defined at $z=0$? Trying to derive Laurent Series of $\csc (1/z)$ to find it.

This question is related to this one. I was able to figure out on my own that the residue of $\displaystyle \sin \left(\frac{1}{z} \right)$ is defined at $z=0$ by finding the Laurent Series of ...
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1answer
14 views

Converting from complex to sinusoidal form and vise versa [closed]

I'm having some trouble understanding this type of transformation. The materials provided by my professor doesn't even mention the method that is being used to switch from complex to sinusoidal and ...
0
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0answers
33 views

Cartesian equation from the complex equation

Find the Cartesian equation for the curve corresponding to the equation $|{z+8\over 16j-z}|=3$ Describe what curve is represented by the equation. Does my answer look correct? $|z+8|=3|16j-z|$ set ...
1
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1answer
19 views

Complex conjugation of fractional powers

I would like to know when the complex conjugate can be moved outside of a real power, i.e. when is it true that $$(\overline{z})^p = \overline{z^p}$$ where $p$ is real? I wrote $z$ in exponential ...
3
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0answers
34 views

Cosine Inequality

Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$ This problem was given in the ...
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0answers
16 views

Method for determining where Laurent series converge

I have to find the Laurent series for $f(z) = \frac{1}{z^2(4z-1)}$. I know there are two series, centered at 0 and at 1/4, because that is where $f$ is not analytic, and I found the series using the ...
0
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4answers
57 views

Prove that a product of two complex numbers has zero imaginary part

This is my homework, which reads as follows: Let $z_1, z_2$ be complex numbers. Prove that when $z_1z_2 \neq -1$ and $|z_1| = |z_2| = 1$, then the imaginary part of $$ \frac{z_1 + z_2}{1 + z_1z_2} $$ ...
1
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1answer
24 views

Does every non-trivial $\mathbb{C}$-algebra contain an element which is not a square?

Let $A$ be a $\mathbb{C}$-algebra. Letting $i$ be the imaginary unit, then for each $a \in A$ we have $$ a = \left(\frac{a-1}{2}\right)^2 + \left(i\frac{a+1}{2}\right)^2 $$ such that each element of ...
2
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0answers
26 views

Graphing/visualizing a complex parametric plot without using mathematica

I am trying to visualize the parametric plot in $\mathbb{C}$ of the curve $\gamma$ defined for $t\in[-\infty,\infty]$ as $$\gamma(t)=\exp\left(-t^{2}+\frac{t}{\sqrt{1+t^2}}i\right).$$ I think I find ...
0
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2answers
15 views

Definition of complex argument: E.g. what is the locus $\arg z \geq 3 \pi/4$?

Say that $\arg z$ has principal values $(-\pi,\pi]$. Then should the locus of points $z$ such that $\arg z \geq 3 \pi/4$ be understood to mean ... Just the region bounded by the rays $\arg z = 3 ...
2
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2answers
162 views

What is $\lfloor i\rfloor$?

So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. ...
0
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2answers
36 views

Among complex $z$ such that $|z-25i|\leq 15$, which have…

Among the complex numbers $z$ which satisfies $|z-25i|\leq 15$, find the complex number $z$ having: (A) Least positive argument (B) Greatest positive argument (C) Least modulus (D) Greatest ...
0
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1answer
34 views

Fractional Exponents Confusion

Let a and b be natural numbers (not including zero). Is it true that will not equal for all possible solutions? For instance, if a=b the would always give an output of x (assuming you don't start ...
0
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1answer
28 views

How to change complex numbers into polar form? [closed]

How do I changecomplex numbers, for example $2+3i$ to polar form of $re^{i\theta}$. Thank you for any answers.
0
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1answer
55 views

Primitive of $\frac{1}{z}$ [closed]

I want show that the function $w: \mathbb{C}^{*} \rightarrow \mathbb{C}$ determine by: $$ w(z) = \frac{1}{z} $$ Hasn't a primitive function defined in $\mathbb{C}^{*}$, I have a primitive function of ...
1
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2answers
22 views

Laurent Series about $z=0$ of $f(z) = \frac{1}{z^3 - iz}$

So far: $$ \frac{1}{z^3 - iz} = \frac{1}{z(z^2 - i)} = \frac{i}{z} - \frac{iz}{z^2 - i} $$ Now I see that: $$ \frac{-iz}{z^2 - i} = z\left(\frac{i}{i - z^2}\right), $$ and this is where I get stuck. ...
0
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1answer
35 views

How to find the derivative of $f(z)$ if $z\in \mathbb C$

How to find the derivative of $f(z)$ if $z\in \mathbb C$ Let $z=x+iy$, then $f(z)=u(x,y)=iv(x,y)$ is it simply $u_x+iv_x=u_y+iv_y$?
2
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1answer
70 views

If $|z_n-z_m|> 2$ for every $n\ne m$ then $\sum \frac{1}{z_n^3}$ converges

Let $(z_n)$ be a sequence of non-zero complex numbers such that $\forall n,m, n\neq m\implies |z_n-z_m|> 2$ Prove that $\sum \frac{1}{z_n^3}$ converges. I'm clueless with this problem. A ...
0
votes
2answers
51 views

Quick Question - Complex Roots of Polynomials?

I'm asked to solve for Z where $$\frac{z+i}{2z-i} = \frac{-1}{2} + i\frac{\sqrt 3}{2}$$ As a result i got $$2z = \sqrt{3}zi + \frac{i}{2} - i^2\frac{\sqrt 3}{2} - i$$ The answer is supposed to be ...
2
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1answer
71 views

How can I express $ i^{2i}$ in the form $x + iy$?

I'm not sure how to begin since this is not in the form $re^{i \theta}$.
2
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0answers
42 views

Multiplying two radicals with negatives, simple algebra? [duplicate]

Evaluate $$ \sqrt{-9}\sqrt{-4} $$ Now, I am told that $\sqrt{a}\sqrt{b}=\sqrt{ab}$, so I should be able to simply write $$ \sqrt{-9}\sqrt{-4} = \sqrt{(-9)(-4)}=\sqrt{36} = 6 $$ However, I am also told ...
0
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2answers
24 views

Finding Possible Meromorphic functions on $\mathbb{C}$

I am trying to find all meromorphic functions on $\mathbb{C}$ such that: $$ \mid f(z) \mid \leq (\frac{3 \mid z \mid}{\mid z + 1 \mid})^{3/2}$$ Can I express the functions as: $$f(z) = ...
1
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1answer
81 views

Proof of an inequality in C ,(2)

Let $n\ge 2$is a integer,$z_{1},z_{2},\cdots,z_{n}$ are $n$ complex numbers Prove that $$\color{crimson}{\sum_{k=1}^{n}|1+z_{k}|+\dfrac{1}{n-1}\sum_{1\le i<j\le ...
0
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0answers
12 views

How can I separate the real and imaginary parts of this Ikeda mapping?

How might I separate the real and imaginary parts of this mapping? So I can plot and compare real curves. $E_{n+1} = A+BE_ne^{i\left|E_n\right|^2}$ where $E_n = x_n+iy_n$.
2
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1answer
28 views

Quick Question - Complex roots of polynomials?

I was asked to find solutions to $z^3 = 1$ and give my answer in Cartesian form. I got $1, -1/2 \pm i\sqrt{3}/2$ (b) Hence solve the equation $(z+i)^3 = (2z-i)^3$ Little help on this one? Any help ...
0
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0answers
12 views

Complex Number Powers of Coprime Rational Powers

I'm trying to figure out $z^{p/q}$ where $p,q$ are coprime. Suppose I want to find $z^{2/7}$ where $z=128$. I can rewrite $z=128e^{0}$ Now I know that the $z^{1/7}$ roots are $2e^{k2\pi i/7}$ for ...
-1
votes
1answer
51 views

Inequality of complex numbers involving modules [duplicate]

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$ My try: I wrote $|z|^2$ as $z\times \bar z$, but I didn't get to any result. Can ...
0
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0answers
18 views

limits and convergence of sequences complex

For the following sequence discuss its limits and whether the convergence is uniform, in the region $\alpha \leq \left | z \right |\leq \beta $, for finite $\alpha$,$\beta >0$. $$\left \{ ...
2
votes
1answer
50 views

Show that $\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}…\cot \frac{(m-1)\pi}{2m}=1$

Prove: $$\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}...\cot \frac{(m-1)\pi}{2m}=1$$ This is a roots of unity problem. I managed to show a similar example for $\cos$ by the ...
0
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0answers
13 views

little agebra help, complex numbers

Can someone please explain this to me, I dont understand how to go from $ [ \psi-1+r( e^{2i\omega} - 4e^{-\omega}+6-4e^{-i\omega} + e^{-2i\omega})] A\psi6{n}e^{i\omega j} $ to this line here. $ ...