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
votes
2answers
63 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. ...
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
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 ...
1
vote
1answer
68 views

Evaluation of Residua

Suppose that I have the following complex valued function, and want to evaluate its residua: $$h(z)=\frac {z^5}{(z-3)(z^4+2)}$$ For both parts of the denominator we will have simple poles. For our ...
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 ...
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 ...
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. ...
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 ...
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} = ...
3
votes
3answers
125 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 ...
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 $(0,i)$.
-4
votes
3answers
96 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
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., ...
1
vote
3answers
75 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 ...
-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 ...
0
votes
0answers
40 views

Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) $$s_{pm} = j \cos \left[ ...
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. ...
3
votes
5answers
276 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 ...
4
votes
3answers
63 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 ...
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
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 ($\mathbb{C}^2$ , $l^1(\{1,2\})$ ...
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
80 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 ...
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 ...
0
votes
3answers
86 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 $|2z-1|+|3z-2|;\,\,z\in\mathbb{C}$ ...
1
vote
0answers
29 views

Find arg of complex function

Find $$Arg \frac{1+z}{1-z}$$. I transfomed it and I have something like that: $$Arg\frac{1-x^2-y^2+2iy}{(1-x)^2+y^2}$$ And do not any idea how to find Arg. I tried to calculate it but it's horrible ...
0
votes
1answer
14 views

Characterizing an anlytical function by its value on the real line

Quote from Strichartz's book "A guide to distribution theory and fourier transform", Chap 4 But an analtic function is determined by its values for z real Why is that so??
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: $ ...
1
vote
0answers
68 views

Can $\zeta(s)$ be written in the form $\zeta(s)=\Re(\zeta(s))+i·\Im(\zeta(s)) $ for some subset of $\mathbb{C}$?

Can $\zeta(s)$ be written in the form $\zeta(s)=f(s)+g(s) i $ for some subset of $\mathbb{C}$? I mean, is it possible to develop at least one of the formulas of $\zeta(s)$ so you get something like... ...
2
votes
2answers
57 views

Find the poles of $f(z)=\frac 1{1+z^w}$ for $w \gt 1$

I am trying to use contour integration on the following integrand between $0$ and $\infty$, however I am not sure how to go about finding the poles for it: $$f(z)=\frac 1{1+z^w},w \in \mathbb Z:w \gt ...
4
votes
1answer
50 views

Iteration of $\log(z) / \sqrt{z}$

The complex function $\log(z) / \sqrt{z}$ is a curiosity that I find interesting since one can express $e^{i\pi}+1=0$ as $\log(-1) / \sqrt{-1} = \pi$. My question is, what is the significance of the ...
1
vote
1answer
27 views

How to practically classify singularities in complex analysis?

I am having trouble developing an intuition around the different types of singularity in complex analysis. The types of singularity that I am aware of are: Poles - These arise at $a_{0}$ when ...
2
votes
1answer
60 views

De Moivre's theorem

Use De Moivre's theorem to show that $$\cos4x=8\sin^4x-8\sin^2x+1$$ Hence show the one of the roots of the equation $8z^4-8z^2+1=0$ is $\sin\frac{\pi}{8}$ and express the other roots in polar form. ...
0
votes
0answers
7 views

negative sign in direction of wave propagation

Say I have a EM wave that goes in the Z direction and E=Eo*exp(-jkz). Why does the negative sign mean the wave travels in the +Z direction and exp(+jkz) means it travels in the -Z direction?
1
vote
0answers
11 views

Lower Bound on Real Part of product?

For complex vectors $u,z,a\in\mathbb{C}^n$, are there any interesting lower bounds on the quantity $$ \Re\left[(a^*z)^2(u^*a)^2\right] $$ in terms of $|a^*z|^2, |u^*a|^2$ that are tighter than $$ ...
0
votes
3answers
30 views

complex number with absolute and algebra

Question if $|x+ i y-1|=|x+ i y-2 i|$ , where $x$ and $y$ are real, express $y$ in terms of $x$ My Solution: +/- $(x+ i y-1)=x+ i y-2 i $ case 1: $(x+ i y-1)=x+i y-2 i $ $-1=-2 i$ (invalid) ...
2
votes
6answers
577 views

solving difficult complex number proving

if $z= x+iy$ where $y \neq 0$ and $1+z^2 \neq 0$, show that the number $w= z/(1+z^2)$ is real only if $|z|=1$ solution : $$1+z^2 = 1+ x^2 - y^2 +2xyi$$ $$(1+ x^2 - y^2 +2xyi)(1+ x^2 - y^2 ...
5
votes
2answers
171 views

A (basic?) contour integration problem

I am trying to prove the following using complex analysis: $$\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{a^{2}+n^{2}}=\frac{\pi}{a\sinh(a\pi)}$$ I am told to use the following function: ...
1
vote
1answer
37 views

Difference between the formula of Roger Cotes and Euler

What was the difference between the formula that Roger cotes derived and that euler got? I mean to say that Euler got the following formula : $$e^{ix} = \cos x+i \sin x$$ And Cotes got the following ...
0
votes
0answers
55 views

Supposing $h$ has a pole, order m, at $z_0$, show the existence of a neighbourhood of $z_0$ and a new complex differentiable function $g$.

Suppose that $h$ has a pole of order $m\in\mathbb{N}$ at the point $z_0$. Show that there exists a neighbourhood of $z_0$ and a complex differentiable function $g$, so that $g(z_0)=z_0$, where ...
1
vote
2answers
33 views

Simplify this complex number [closed]

Simplify this complex number modulus $$\left|\left(\frac{100\pi i}{100+100\pi i}\right)\left(3\sqrt{2}\right)\right|$$
2
votes
2answers
82 views

Calculate the residue of this function

Find the residue at $z=0$ of the function $f(z)=\frac{\cot z}{z^4}$ I know that $z_0=0$ is a pole of order $k=5$, and $$Res(f;z_0)=\frac{\phi(z_0)^{(k-1)}}{(k-1)!}$$ but I cannot get the right ...
1
vote
2answers
40 views

The magnitude of the complex number

How can we find $|M|^2$ for $$M=\frac{e^{2(1+{\rm i})l}-e^{-2(1+{\rm i})l}}{e^{2(1+{\rm i})x}-e^{-2(1+{\rm i})x}} ?$$ We have $$M=\frac{\cos 2l[e^{2l}-e^{-2l}]+{\rm i} \sin 2l[e^{2l}+e^{-2l}]}{\cos ...
1
vote
3answers
55 views

Complex Integration, residues

Evaluate the following integrals by the method of residues i)$\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$, a real ii)$\int_0^\infty \frac{x^\frac{1}{3}}{1+x^2}dx$ I'm a little lost to ...
4
votes
1answer
66 views

Simplify $\sqrt{-3}$

I was reading about this known fallacy $$ -1 = i^2 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 $$ and according to Wikipedia "The fallacy is that the rule $\sqrt{xy} = ...