# Tagged Questions

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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### 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? ...
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### 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.
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### 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 ...
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### Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
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### Can there be a numerical system in which logarithms can be expressed in terms of exponents in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
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### 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 ...
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### Solve $(x+iy)^2=8+6i$, for $x$ and $y$

Given $(x+iy)^2 = 8+6i$, find the values of $x$ and $y$. Hence find $\sqrt{8+6i}$. My question is when we solve we get $x = 3$ and $x = -3$, which give and $y = 1$ and $y = -1$ then Why $-3-i$ is ...
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### 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 ...
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### 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, ...
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### Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
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### 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. ...
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### Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $z_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the ...
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### Cosine Inequality, Geometric interpretation in the complex plane

The following identity was given as an exercise in the course notes for a complex analysis course. I am able to solve it (the proof is given below), but am unsure of the geometric interpretation of ...
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### 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 ...
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### What does it mean to divide a complex number by another complex number?

Suppose I have: $w=2+3i$ and $x=1+2i$. What does it really mean to divide $w$ by $x$? EDIT: I am sorry that I did not tell my question precisely. (What you all told me turned out to be already known ...
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### 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 ...
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### 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 $p > 0$ ...
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### 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 ...
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### 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 $$\sum_{k=1}^{n}|1+z_{k}|+\dfrac{1}{n-1}\sum_{1\le i<j\le n}|1+z_{i}z_{j}|\ge\sum_{k=1}^{n}|z_{k}|$$ for ...
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### 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 ...
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### Prove that the point will go 3 times around ellipse

I'd like to prove that if a point $z$ goes once around ellipse with focus $2,-2$ then point $z^3-3z$ goes 3 times around some ellipse with the same focus. I was thinking (since ellipse is a set of ...
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### 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 $E_n = x_n+iy_n$.
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### Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
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 ...
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 ...