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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0
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2answers
44 views

Finding the complex roots of an equation.

I feel ridiculous asking this, its something I should be able to do, however I shall ask anyway. I am doing a calculation that requires me to find the roots of the equation ...
0
votes
1answer
30 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...
0
votes
0answers
43 views

Primitive root of unity in complex plane

I have a polynomial $p(x) = -3x^{6}+ 4x^{5}-x^{4}-3x^{2} +6x-1$ in a complex plane and I need to transform it with DFT. Based on the degree of the polynomial makes ...
5
votes
2answers
216 views

Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if the Riemann Hypothesis is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...
2
votes
1answer
43 views

question involving remainder of complex function

The question says - Dividing $f(z)$ by $(z-i)$, we get remainder $i$ and dividing by $z+i$, we get remainder $1+i$. Find the remainder upon division of $f(z)$ by $z^2 + 1$ How do I go about ...
3
votes
1answer
39 views

Solving characteristic equation to find eigenvalue.

I came across the following question: The characteristic polynomial of a $3 \times 3$ matrix $A$ is $|\lambda I -A| = \lambda^3 + 3 \lambda^2+4 \lambda +3$. Find $trace(A)$ and $det(A)$. I know ...
0
votes
0answers
16 views

Comparing the supremum of Maclaurian series with the function.

Suppose $f$ is an entire funciton with the Maclaurin Series $$a_0+a_1z+a_2z^2+\cdots $$ Show that if $r>0$ then $$|a_0|^2+|a_1|^2|r|^2+|a_2|^2|r|^4+|a_3|^2|r|^6+\cdots < \sup_{|z|=r} |f(z)|^2 ...
-1
votes
1answer
55 views

What are hidden facts of Complex number? [duplicate]

I want to know how complex number can be used in real life. What are hidden usage of complex number in real life. Can anyone explain ? Thank you !
0
votes
1answer
19 views

Residues and poles show that

Show that i) $\displaystyle\operatorname{Res}\limits_{z=\pi i}\frac{z-\sinh z}{z^2\sinh z}=\frac{i}{\pi}$ ii) $\displaystyle\operatorname{Res}\limits_{z=\pi i}\frac{\exp(zt)}{\sinh z}+ ...
0
votes
1answer
37 views

Solving a complex number inequality involving absolute values.

Here is the relevant paragraph (from "Complex numbers from A to Z" by Titu Andreescu and Dorin Andrica) : Original question : How does $\left | 1+z \right |=t$ imply $\left | 1-z+z^2 \right ...
1
vote
2answers
44 views

Can someone explain in simple terms how to understand and calculate n to the power of i for n > 1? [closed]

I know how to add, subtract, multiply and divide any number by a complex number, but it is mysterious how one go about calculating $2$ to the power of $i$ for example. I would like to understand from ...
7
votes
2answers
74 views

Imaginary $\cos^{-1}$ value significance?

When I was bored in AP Psych last year, I jokingly asked myself if there was a cosine inverse of $2$. Curious about it, I tried calculating it as follows: $$ \begin{align*} \cos (x) &= 2 \\ \sin ...
1
vote
1answer
14 views

Studying electronic filters; how do I've to find the following complex argument limits?

$$\lim_{\omega\rightarrow0} \left(\arg\left(\frac{a+b+\left(i\omega l\right)+\left(\frac{1}{i\omega c}\right)}{a+b+f+g+\left(i\omega l\right)+\left(i\omega L\right)+\left(\frac{1}{i\omega ...
3
votes
3answers
87 views

Is $|z+i| = |z-1|$ a circle with radius $\sqrt{1^2+1^2=1}$ and origin $(1,-i)$?

Is $|z+i| = |z-1|$ a circle with radius $\sqrt{1^2+1^2}=1$ and origin $(1,-i)$? Because I know $|z+i| = 3$ is is a circle with radius $3$ and origin $(0,i)$.
5
votes
6answers
115 views

Proof of Euler's formula that doesn't use differentiation?

So I saw a 'proof' of the sine and cosine angle addition formulae, i.e. $\sin(x+y)=\sin x\cos y+\cos x \sin y$, using Euler's formula, $e^{ix}=\cos x+i\sin x$. By multiplying by $e^{iy}$, you can get ...
1
vote
2answers
27 views

Solving inequalities on both sides with complex numbers

I need to sketch this region $\left \{ z\in\mathbb{C}| |z-i|\leq |z-1| \right \}$. I'd like some assistance with solving this inequality because I think that's where I'm going wrong. To solve the ...
-4
votes
3answers
94 views

1 is equal to -i [duplicate]

Pretty simple, but I'm sure there's some subtlety to it I'm missing. $$(-i)^2=1 \Rightarrow \sqrt{1}=-i \Rightarrow 1=-i$$ Looking at an Argand diagram however, gives some reason to doubt this. ...
1
vote
1answer
37 views

Evaluating residua and simplifying complex expressions.

My question is in two parts, so please forgive its long-winded nature. Lets say that I want to find the residua of the following complex function: $$f(w)=\frac{2w+1}{w(w^3-5)}$$ Let us, ...
2
votes
2answers
23 views

Showing internal angles of a square are unaffected by a mapping

I recently had an exam in complex analysis, and I am slightly confused by one of the questions, so I'd appreciate any clarification: The mapping from the complex $z$ plane to the complex $w$ plane ...
1
vote
1answer
73 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the ...
3
votes
6answers
1k views

For a polynomial $p(z)$ with real coefficients if $z$ is a solution then so is $\bar{z}$

I can "see" it intuitively, though I do not know how correct this is: in a complex conjugate we change the sign of all imaginary parts, and since the effect of all imaginary parts cancels out on the ...
0
votes
2answers
61 views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ ...
1
vote
0answers
52 views

complex rank-one update

I'm trying to find the eigendecomposition of a rank-one update to a complex matrix $D + uv^T$. The matrix $D$ is diagonal, but not the identity. It has unique imaginary entries along the diagonal. ...
0
votes
2answers
37 views

Rotations/Transformations with Complex Numbers/Eulers Formula

Hello, I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral ...
3
votes
1answer
65 views

If $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real, then so are a,b,c

Let $a,b,c$ be complex numbers with distinct magnitudes such that $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real. Prove that $a,b,c$ are real numbers as well. I tried to go for ...
0
votes
1answer
25 views

Center and angle of complex function

Does a complex function of type $f(z)=az+b $ always have a center and angle (of rotation) or only when $b=0$ since $b\neq0$ represents a translation?
1
vote
2answers
50 views

Finding the orthogonal complement of a complex subspace

Let $i := \sqrt{-1}$ . Consider $W \subseteq \mathbb{C}^3$ defined by $W := \{(1, 0, i),(1, 2, 1)\} $. Find $W^\perp$. My biggest issue with this problem is not knowing how to extend the basis of $W$ ...
2
votes
1answer
56 views

Find the poles/residues of $f(z)=\frac{\sin(z)}{z^4}$

I'm trying to calculate the residua of the following complex function but am encountering problems trying to determine its poles: $$f(z)=\frac{\sin(z)}{z^4}$$ Expanding the denominator shows that we ...
2
votes
2answers
84 views

How to prove that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable (if it is true)?

Can someone show me: If $x$ is a real number, then $\cos^2(x)+\sin^2(x)= 1$. Is it true that $\cos^2(z)+\sin^2(z)=1$, where $z$ is a complex variable? Note :look [this ] in wolfram alpha ...
2
votes
3answers
60 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. ...
1
vote
1answer
30 views

prove that $\Bbb Z/n \Bbb Z \cong \mu_n$

I need to prove that $\Bbb Z/n \Bbb Z \cong \mu_n$ $\Bbb C^x \gt \mu_n = \{z \in \Bbb C^x | z^n = 1 \}$ what i tried - I tried building a homomorphism $f: \Bbb Z \to \mu_n$ such that $f(z) = e^{{2 ...
1
vote
2answers
41 views

All solutions of $z\in \Bbb C, \cos z = i$

I want to find all solutions to $\cos z = i$ Okay so $$\cos z = \frac12(e^{-iz}+e^{iz})=i$$ $$e^{-iz}+e^{iz}=2i$$ $$e^{-iz}+e^{iz}=2e^{\frac\pi 2i + 2\pi i n},n\in \Bbb Z$$ $$1+e^{2iz} = ...
1
vote
3answers
74 views

System of equations with complex numbers-circles

The system of equations \begin{align*} |z - 2 - 2i| &= \sqrt{23}, \\ |z - 8 - 5i| &= \sqrt{38} \end{align*} has two solutions $z_1$ and $z_2$ in complex numbers. Find $(z_1 + z_2)/2$. So far ...
3
votes
3answers
122 views

Why is $\sqrt{-i} \neq i\sqrt{i}$?

I wanted to figure out the square root of $-i$. Since $\sqrt{-x} = i\sqrt{x}$, $\sqrt{-i}$ should equal $i\sqrt {i}$, however, WolframAlpha said it was false. However, if I do say that ...
4
votes
3answers
62 views

How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
1
vote
1answer
42 views

Analytic function for which $\overline{f(z)} \neq f(\overline{z})$?

Since $\overline{f(z)} = f(\overline{z})$, where $\overline{z}$ denotes the complex conjugate of $z$, already works for polynomials with coefficients in $\mathbb{R}$, the exponential function, etc., ...
3
votes
5answers
273 views

Evaluate the complex integral of function

Use the residue theorem to evaluate $\int_\gamma \frac{z^5}{1-z^3}dz$ where $\gamma$ is the circle $|z|=2$. I have that $z_0=1$ is a singularity point and taking $g(z)=z^5$ and $h(z)=1-z^3$ and ...
-2
votes
2answers
42 views

please help me to solve $\frac{1-\exp(-10j\pi)}{1-\exp(-2j\pi)}$

Why the result of the below statement is equal to $5$? $$\frac{1-\exp(-10j\pi)}{1-\exp(-2j\pi)}$$ I compute this way and it results NaN! $1-\exp(-j \cdot 10 \cdot \pi)= 1-(\cos(10 \cdot \pi)-j ...
3
votes
1answer
38 views

Distinguishing Primitive vs. Nonprimitive Roots of Unity

In a question here, the solution given states that $$\zeta=\cos{(\pi/8)}+i\sin{(\pi/8)}$$ is a primitive 8th root of unity. I was under the impression that the primitive roots of unity were given my ...
1
vote
1answer
33 views

How can I find a norm of a linear transformation $T(x,y) = (ax+cy, bx+dy)$?

Let a linear transformation $T : \mathbb{C}^2 \to \mathbb{C}^2$ s.t $T(x,y) = (ax+cy, bx+dy)$ where $a,b,c,d \in \mathbb{C}$. Now, find the norm of T equipped with ($\mathbb{C}^2$ , $l^1(\{1,2\})$ ...
1
vote
1answer
27 views

Exponential function and residues

Show that singular point of function $f(z)=\frac{1-e^{2z}}{z^4}$ is a pole and find the order $m$ of that pole and the corresponding residue. Here my question begins with the singular point ...
2
votes
1answer
29 views

Singular points and residues

In each case write the principal part of the function at its isolated singular point and determine whether that point is a pole, a removable singular point, or an essential singular point. ...
0
votes
2answers
30 views

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior. my works I'm stuck in next step
1
vote
4answers
71 views

Definition of exponential function, single-valued or multi-valued?

If we define $$e^z=1+z+\frac{z^2}{2!}+\cdots$$ then it is single-valued. However, if we write $$e^z=e^{z\ln e}$$ then it is multi-valued. Besides, $a^z$ is multi-valued in general. It is kind of ...
2
votes
3answers
52 views

Complex numbers: $|\frac{1}{x}-\frac{1}{y}| = \frac{|x-y|}{|x||y|}$?

I need the result for a proof, but I can't seem to verify it. If $x,y$ are distinct nonzero complex numbers, why is it true that $|\frac{1}{x}-\frac{1}{y}| = \frac{|x-y|}{|x||y|}$? Starting with the ...
1
vote
1answer
42 views

Expressing $\frac {\sin(5x)}{\sin(x)}$ in powers of $\cos(x)$ using complex numbers

Am I correct in thinking that if $z_1=a+ib$ and $z_2=c+id$, then it is not generally true that $$ \frac {\textrm{Im}(z_1)}{\textrm{Im}(z_2)} = \textrm{Im}\left(\frac {z_1}{z_2}\right) $$ I did a ...
3
votes
2answers
79 views

How to integrate $e^{-\cos(\theta)}\cos(\theta + \sin(\theta))$

I am struggling to find a way to evaluate the following real integral: $$\int_{0}^{2\pi}e^{-\cos(\theta)}\cos(\theta + \sin(\theta))\:\mathrm{d}\theta$$ The exercise started by asking me to ...
0
votes
3answers
85 views

What's wrong with this?

What's wrong with this : $$e^{i\pi} = -1$$ $$\therefore e^{2i\pi} = 1$$ $$\therefore log \left( e^{2i\pi} \right ) = log(1) = 0$$ $$\therefore 2i\pi = 0$$
1
vote
2answers
46 views

Find minimum value of $|2z-1|+|3z-2|;\,\,z\in\mathbb{C}$?

I tried this question using many different ways (triangle inequality, geometric interpretation, etc) but I didn't get the correct answer. The minimum value of $|2z-1|+|3z-2|;\,\,z\in\mathbb{C}$ ...
0
votes
4answers
136 views

How to solve $e^{ix}=i$?

This is a question related to another posted question: The answer to the following question "Find all solutions to: $e^{ix}=i$" is as follows: "Euler's formula: $e^{ix}=\cos(x)+i\sin(x)$, so: $ ...