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
1answer
70 views

What is the Geometric interpretation of $i^i$?

We know that $i^i$ is real. But how to explain it geometrically maybe in terms of rotation. like we can explain geometrically multiplication of two complex numbers and so on. Can someone show me a ...
0
votes
1answer
32 views

Proper definition of concyclic?

Let $z_1,z_2...,z_n$ be points in the complex plane, then if there exists $Z$ such that $$\vert Z-z_k\vert=a\in\{\text{Real Numbers}\}$$for all $k\in \{1, 2, 3...,n\}$, then $z_1,z_2...,z_n$ are ...
1
vote
1answer
56 views

Finding point where angular bisector meets circumcircle in complex plane

If $A(z_1)$,$b(z_2)$ and $C(z_3)$ are vertices of a triangle. It is inscribed in circle |z|=2. If internal angular bisector of A meets the circumcircle at $D(z_4)$. Find $z_4$ interms of $z_1$,$z_2$ ...
-1
votes
2answers
130 views

Euler's formula for off-center circle [closed]

A circle with radius $R$ and center at $(a,b)$ is given by the formula $(x-a)^2 +(y-b)^2 = R^2$. A circle with radius $R$ whose center is at the origin is given by Euler's formula: $R e^{i \theta}$. ...
1
vote
2answers
47 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
3
votes
2answers
69 views

How to solve this system of equations for $x^2+y^2+z^2$?

For the complex numbers $x,y,z$, the system of equations $x^2-yz=i~~~~~ y^2-zx=i~~~~~ z^2-xy=i$ It is not easy for me to get $x^2+y^2+z^2$ from the above. I don't need the values of $x,y,z$ I'm ...
5
votes
1answer
107 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
14
votes
2answers
1k views

Why does the boundary of the Mandelbrot set contain a cardioid?

In a comment to a previous answer it has been mentioned that the boundary of the Mandelbrot set contains the cardioid $$ c = e^{it} \, \frac{2 - e^{it}}{4} $$ but how can we prove this?
1
vote
2answers
33 views

complex numbers equation, find all z…

So i have to find all $z\in \mathbb{C}$ that solve these two equations(separately) first: $\bar{z}+z=i(\bar{z}-z)$ second: $\bar{z}+z^n=i(\bar{z}-z^n), \forall n \in\mathbb{N}$ So basically, i ...
5
votes
3answers
66 views

Infimum taken over $\lambda$ in $\mathbb{C}$

I want to calculate the infimum of $$ |\lambda-2|^2+|2\lambda-1|^2+|\lambda|^2 $$ over $\lambda\in\mathbb{C}.$ I choose $\lambda=2,1/2,0$ so that one term in the above expression becomes zeros and ...
1
vote
1answer
49 views

Series involving complex roots

$$ \frac{1}{2-a_1} + \frac{1}{2-a_2} + \dots + \frac{1}{2-a_{n-1}} = \frac{(n-2)2^{n-1}+1}{2^n - 1} $$ Here $1,a_1,a_2,\dots,a_{n-1}$ are $n$-th roots of unity I know the sum of roots is 0. I think ...
4
votes
4answers
229 views

Proof: Derivative of $(-1)^{x}$

The derivative for $(-1)^{x}$ is \begin{equation} \frac d{dx}\left[(-1)^x\right]=i\pi(-1)^{x} \end{equation} But why? What happens with higher order derivatives? Thanks in advance.
4
votes
1answer
109 views

Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$

Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$. If $z=x+iy$ then $$\frac{z+2}{2-i}=\frac{x+2+iy}{2-i}\times \...
3
votes
1answer
226 views

Why is '1' the multiplicative identity of complex numbers and quaternions?

I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me ...
7
votes
5answers
125 views

$e^{i\theta}$ versus $\cos\theta+i\sin\theta$

I am teaching an basic university maths course, and have been thinking about the complex numbers part. Specifically, I was wondering why I should include Euler's formula in my course. This led me to ...
1
vote
1answer
80 views

Separate real and imaginary part of $j \cos (z)$

Given the following expression $$w = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right] = j \cos (z)$$ (which is related to this question; $n,...
1
vote
1answer
106 views

Question regarding complex numbers and real numbers?

I have two questions... If we take $(-1/3)^{(-1/3)}$ it would equal $-1.44224957$ since... $$(-1/3)^{-1/3}$$ $$\frac{1}{(-1/3)^{(1/3)}}$$ $$\frac{1}{-0.6933612744}$$ $$-1.44224957\ldots$$ Yet when I ...
1
vote
4answers
206 views

Find all complex numbers $z=a+bi$ such that $z^3=8$.

Find all complex numbers $z=a+bi$ such that $z^3=8$. I'll be happy if someone say me with what steps I have to start solving this problem.
0
votes
8answers
88 views

Trigonometric Property

How can I show that the following property holds? $2\cos(4a)+2\cos(2a)+1=\displaystyle\frac{\sin(5a)}{\sin(a)}$ I've been trying to derive it to no avail. What would be a way to approach similar ...
2
votes
2answers
101 views

Representation of Heaviside function's Fourier transform

I've seen here that the Fourier transform of Heaviside function $\Theta(t)$ is $$ \Theta(\omega) = \frac{1}{i\omega} + \pi \delta(\omega) \tag{1}$$ But in some physics texts and here I've seen the ...
2
votes
2answers
33 views

Property of polynomials proof

Let$$P(z)=\sum_{k=0}^n a_kz^k=a_0+a_1z+...+a_nz^n$$ be an N-th degree polynomial of a complex variable z, where the $a_k$ are complex constants. Now,$$\vert a_0\vert-\vert a_1\vert x-...-\vert a_n\...
2
votes
0answers
61 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
2
votes
2answers
53 views

Defining set of interior points of a triangle

Is there a way, given that $z_1,z_2 \ \text{and} \ z_3$ are the vertices of a triangle in the complex plane, to characterize all point that are inside of the triangle?
1
vote
3answers
120 views

Where's the mistake in this calculation? [duplicate]

Obviously something is wrong with this, but where is the error and why is it one? $$ \begin{align} \sqrt{-1} &= (-1)^{1/2} \\ &= (-1)^{2/4} \\ &= \sqrt[4]{(-1)^2} \\ &= \sqrt[4]{1} \\ ...
7
votes
2answers
170 views

proof for $\frac{1}{i} = -i$?

My physical chemistry textbook seems to be making the implicit assumption that $\cfrac{1}{i} = -i$. I'm not sure how this is valid. Here is the snippet of relevant steps: $\cfrac{i\hbar}{f(t)}$$\...
7
votes
5answers
413 views

Definite integral of even powers of Cosine.

I'm looking for a step-by-step solution to the following integral, in terms of n$$\int_0^{\frac{\pi}{2}} \cos^{2n}(x) \ {dx}$$I actually KNOW that the solution is$${\frac{\pi}{2}} \prod_{k=1}^n \frac{...
1
vote
0answers
260 views

Moving the absolute value inside of an integral involving a complex function

I have the following integral to evaluate $\lvert \int_0^\frac{\pi}{4}e^{iR^2e^{i2\theta}}iRe^{i\theta}d\theta\rvert$ and I want to put the absolute value sign inside of the integral so that I can ...
1
vote
4answers
38 views

Product of roots of unity using e^xi

Find the product of the $n\ n^{th}$ roots of 1 in terms of n. The answer is $(-1)^{n+1}$ but why? Prove using e^xi notation please!
2
votes
1answer
28 views

Finding the residue, $z=n\pi$, and $e^{n\pi}$

I have reached the following point in a residue calculation and am now unsure what to do: $$Res_{z= n\pi}=\lim_{z\to n\pi}\{(z-n\pi)\frac{ e^z}{\sin(z) } \}$$ $$=\lim_{z\to n\pi}{\{\frac{e^z(z-n\pi)+...
8
votes
1answer
134 views

Cosh and Sinh analogs

We know that $$\cosh{x}+\sinh{x}=e^x$$ and that his can be expressed as $$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$ and this works out nicely because the $e^{...
2
votes
3answers
65 views

Complex integration with trigonometric and logarithm

Show that $\int_0^{2\pi}\log\sin^22\theta dx=4\int_0^\pi\log\sin \theta d\theta=-4\pi \log2$ I did $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ taking ...
2
votes
1answer
56 views

Complex sum using Laurent series?

By considering $f(z)=exp(z-\frac{1}{z})$ show that $$ \frac{1}{2\pi}\int_{0}^{2\pi}cos(n\theta-2sin\theta)d\theta=\sum_0^{\infty}\frac{(-1)^k}{k!(n+k)!}\ \forall n\ge1$$ f is holomorphic in $\...
3
votes
1answer
89 views

Is the following function a constant function

Suppose that $f: \mathbb{C} \rightarrow \mathbb{C}$ is entire and bounded on the set $\{z \in \mathbb{C}; Re(z) \leq 0\}$. Is $f$ a constant function. I know by Picards theorem that a non-constant ...
6
votes
3answers
369 views

Entire function with uncountably many zeros

Suppose that an entire function $f$ has uncountably many zeros. Is it true that $f=0$? I have no idea how to proceed with this. Perhaps some theorem that I am not aware of. I have done an ...
4
votes
2answers
221 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did $$\int_0^{\frac{\pi}{2}}...
0
votes
1answer
25 views

About the inequality $\frac{1}{2}>|\frac{z}{c}|, \forall z\in K.$

Let $c\geq 2 diam(K)$, where $K$ is compact in $\mathbb C$. Show that $\frac{1}{2}>|\frac{z}{c}|, \forall z\in K.$
0
votes
2answers
158 views

Trigonometric identities — a parallel RLC circuit connected to an AC-supply [closed]

An RLC-circuit is connected to an AC-supply as in the figure below. $I_{tot}(t)=I_0sin(\omega t+\phi)$ (denoted as $I_{ges} ( t)$ in the picture), $\phi$ is the phase angle between $V_{tot}...
1
vote
1answer
26 views

Showing there exists a complex differentiable function $g$ satisfying $g(z_0)=z_0$, with $g'(z_0) \neq 0$ and that $h(g(z))=(z−z_0)^{−m}$.

This is a follow up to a previous question: (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 $g$.) I'm trying to ...
1
vote
3answers
40 views

If real numbers $x$ and $y$ satisfy the equation $\frac {2x+i}{y+i}= \frac {1+i\sin{\alpha}}{1-i\sin{3\alpha}}$ then quotient $\frac xy$ is equal to?

If real numbers $x$ and $y$ satisfy the equation $\frac {2x+i}{y+i}= \frac {1+i\sin{\alpha}}{1-i\sin{3\alpha}}$, then quotient $\frac xy$ is equal to? Other conditions are ($\alpha \neq k\pi,\ \...
3
votes
1answer
69 views

why is the integer power of a complex number not multi-valued too?

my textbook [H. A. Priestley - Introduction to Complex Analysis] states about the argument of a complex number raised to a power : 'Only when $\alpha$ is an integer does $[z^{\alpha}]$not produce ...
2
votes
2answers
64 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow z^4=-1=e^{i\...
1
vote
2answers
56 views

What does 𝔍(z) mean?

In complex analysis, if z is complex number, what does 𝔍(z) mean? The symbol is a mathematical fraktur capital J, unicode U+1D50D.
1
vote
3answers
72 views

Shortcut Technique for finding Raised Binomials with Imaginary Numbers

Find the Value of $(1+i)^5$ where $i$ is an imaginary number. The answer is $-4\cdot (1+i)$ We can always multiply them manually; but $i$ was wondering if there are any math tricks to quickly ...
0
votes
1answer
57 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 $$\int_{-\infty}^\infty\frac{dx}{x^2+1}+\...
0
votes
2answers
61 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 $\frac{1}{4}(z^{-2}+2z-z^2)$...
1
vote
1answer
41 views

Improper integrals and residues

I'm already read Conway, Churchill and Marsden but I'm still with doubts when it comes to improper integrals. Where come from this relation $$\lim_{R\rightarrow\infty}\int_{-R}^Rf(x)dx+\int_{C_R}f(z)...
0
votes
0answers
52 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 ...
2
votes
1answer
53 views

Question on construction of entire functions

Suppose that $x_i$ and $y_i$ are sequences in $\mathbb{C}$. Can you construct a non constant entire function such that $f(x_i)=y_i$? What happens if $x_i$ have an accumulation point? or what happens ...
3
votes
1answer
57 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 ...
2
votes
2answers
134 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 ...