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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2
votes
2answers
112 views

Show that there is no analytic bijection from the unit disc to $\mathbb{C}$

Show that there is no analytic bijection from the unit disc to $\mathbb{C}$. I am quite unsure how to proceed here. I know for a fact that there is no analytic function from $\mathbb{C}$ to the open ...
-1
votes
1answer
84 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
0answers
28 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 ...
0
votes
1answer
21 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}+ ...
1
vote
2answers
53 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 ...
0
votes
1answer
79 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 ...
2
votes
1answer
46 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 ...
0
votes
1answer
18 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 ...
1
vote
2answers
76 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 ...
7
votes
2answers
85 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 ...
6
votes
6answers
195 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 ...
2
votes
2answers
24 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
84 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 ...
0
votes
2answers
197 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
75 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
41 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
120 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
69 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
690 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
36 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
68 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
69 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
82 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
144 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
103 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
107 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
50 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
86 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
52 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
52 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
65 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
116 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
164 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
383 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
4answers
148 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
40 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
44 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
55 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
69 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
237 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
275 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 ...
1
vote
3answers
103 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
61 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
48 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
18 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
242 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
81 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
64 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
64 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
78 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 ...