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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-3
votes
2answers
26 views

Difference in distance of moving point in complex plane. [closed]

A point moving in the complex plane satisfies the following relation : $$z^2 + \bar{z}^2 = 8$$ What is the difference of the distances of the moving point from $(2\sqrt{2},0)$ and $(-2\sqrt{2},0)$ ...
0
votes
0answers
55 views

Proof that set is a convex set

Let $\displaystyle W(A):=\left\{ \frac{x^*Ax}{x^*x}: x\in C^{n} \right\}$ and $A\in C^{n} $ so that $ A^{*}A=AA^{*} $ $A^{*} $ is the complex conjugate of $A^T$ (the transpose of a) (See: ...
1
vote
0answers
56 views

Unitary Complex Matrix

Let $M$ be a complex matrix $$M:=\begin{bmatrix} 0 & i & -i \\ i & 0 & 1 \\ -i & 1 & 0 \\ \end{bmatrix}$$ 1) Find a unitary complex matrix $Q$ ...
0
votes
0answers
27 views

Area of a triangle on an Argand diagram

I am working on two problems: 1) Find three distinct roots of the equation $8z^3 + 27 = 0$ I solved this and ended up with \begin{align*}z_1 &= \frac32 \left( \cos (\pi/3) + ...
0
votes
1answer
34 views

How can I solve $\sum_{n=1}^\infty\frac{(-1)^{n+1} \cdot z^n}{n}$? [closed]

The answer is given. It is equal to $\ln(1 + z)$. Can you help me solve it?
2
votes
1answer
25 views

Value that a complex number can take

Question: $z_1,z_2$ are two complex numbers with $z_2 \neq 0$ and $z_1 \neq z_2$ and satisfying: $$\left | \frac{z_1 + z_2}{z_1 - z_2}\right | = 1$$ Then $z_1\over z_2$ is: A) Real and negative B) ...
-1
votes
1answer
48 views
0
votes
2answers
28 views

Transformations of points in the plane

Hopefully somebody understands what I mean here, If take a polynomial with complex numbers as input, then I will get a complex number as an output. If the input and output are plotted on an Argand ...
1
vote
1answer
24 views

Inequalities involving the Roots of unity

Let $\epsilon \ne 1$ be a nth root of unity. Then prove that, $\bullet \ |1-\epsilon|\ge\frac{2}{n-1}$ $\bullet |\sin\frac{k}{n}| \ge \frac{1}{n-1}$ Here are my solutions: $\bullet$ To prove ...
5
votes
1answer
39 views

Evaluate the improper integral with residues.

$$\int_0^{\infty} \frac{x^2+1}{x^4+1}dx$$ What i've found are the singularities at: $e^{\pi/4+\pi/2n}$ for $n=0,1,2,3$. But i'm unsure how to calculate the integral since I don't want to include ...
2
votes
1answer
41 views

Determine behaviour of $\sum_{n=1}^{\infty} \frac{1}{n}z^n$ for $|z|=1$

I am trying to analyze behaviour of the series $\sum_{n=1}^{\infty}\frac{1}{n}z^n$ at the set $\{|z| \leq R\}$ where $R$ is the radius of convergence of the series. Since $\lim_{n \to \infty} ...
6
votes
4answers
120 views

What is the most useful/intuitive way to define $\Bbb{C}$, the complex set?

I was wondering whether there is a more common or 'best' way to define/think about the set of all complex number in complex analysis? The first way I can think of is: $\Bbb{C} \triangleq \Bbb{R} ...
0
votes
3answers
60 views

How to solve $\sum_{n=1}^\infty\frac{z^n}{n}$? [closed]

I already know that it is equal to $\ln(1-z)$, but I need to prove that.
0
votes
1answer
16 views

Branch points+cuts, $(z+k)^{1/2}$ and $(z+k)^{-1/2}$

Do the complex functions $f(z)=(z+k)^{1/2}$ and $f(z)=(z+k)^{-1/2}$ have the same branch points? If so, why? Also, would this mean that we can take the same branch cut for both functions? Thanks.
0
votes
1answer
31 views

Complex Numbers: Im having a problem solving this.

I've been working some problems solving and simplifying basic complex numbers. I'm really stuck at this one; I haven't an idea where to begin solving. (I'm a 1st year engineering student doing ...
3
votes
1answer
42 views

Radius of convergence of two power series

I am trying to find the radius of convergence and trying to figure out the behaviour on the frontier of the disk of convergence of the following power series: a) $\sum_{n=1}^{\infty} ...
1
vote
1answer
27 views

Analyticity and differentiability of complex functions

I understand what analytic functions are and what differentiability of a complex function means but I have been reading "advanced engineering mathematics by kreyszig" and it says that the concept of ...
1
vote
1answer
28 views

What is the number of ordered pairs of real numbers (a,b) such that …

Problem : What is the number of ordered pairs of real numbers (a,b) such that $(a+ib)^{2002}=a-bi$ My approach : Multiplying both sides by a+ib we get $(a+ib)^{2003} = a^2+b^2$ $\Rightarrow ...
1
vote
1answer
23 views

Hurwitz Zeta in terms of Bernoulli polynomials.

@Raymond Manzoni showed nicely in this post how the Riemann zeta function is related to the Bernoulli numbers using the Euler-Maclaurin sum. The result is : \begin{eqnarray} \zeta(1-k) = ...
0
votes
1answer
23 views

Find the real and imaginary parts of $f(z)=\frac{2}{z+1}$

Find the real and imaginary parts of $f(z)=\frac{2}{z+i}$. Denote the conjugate of $z$ as $\bar{z}$, then ...
2
votes
1answer
34 views

solve $\sin(z)=-1$ in the set of complex numbers

I'm pretty sure I'm on the right track, but am I missing anything? Can anything further be done with this? Solve $\sin(z)=-1$ in the set of complex numbers. $\sin(z)=-1$ $\Rightarrow{e^{iz}-e^{-iz} ...
1
vote
2answers
55 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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
1answer
19 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
votes
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
vote
2answers
20 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
votes
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
votes
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
votes
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
votes
2answers
37 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
votes
1answer
67 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
votes
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
votes
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
votes
0answers
45 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
votes
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
votes
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
votes
1answer
34 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 ...
-1
votes
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
votes
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
vote
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
votes
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 ...
1
vote
0answers
14 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
votes
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
vote
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
votes
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 ...