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
78 views

Why was $i$ introduced to satisfy this $\sqrt{-1}$?

Can someone explain to me why $$\sqrt{-1} = i$$ I love math and I'm looking at doing it to higher levels. I know that we can NEVER have a square root of a negative number as per my reading hence if I ...
1
vote
2answers
35 views

Complex numbers - roots of unity

Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 - \omega^2} + \frac{\omega^2}{1 - \omega^4} + \frac{\omega^3}{1 - \omega} + \frac{\omega^4}{1 ...
0
votes
0answers
19 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
0
votes
2answers
27 views

Determine the locus of a equation Quickly[Mental Math]

if $Z=X+iy$ then determine the locus of the equation $\left | 2Z-1 \right | = \left | Z-2 \right |$.I can tell that it a circle equation and it is $x^2 + y^2 = 1$.There are a lot of equation in my ...
2
votes
1answer
15 views

What does s=jω actually mean in terms of the complex plane and Laplace transforms?

I was trying to solve a problem on RC circuits. The current source was of the form $\cos (\omega t)$ which transforms in the manner of Laplace to $\frac{s}{s^2+\omega^2}$. I thought I’d use the ...
1
vote
2answers
32 views

Solve complex exponential equation

I need to solve an expression of this kind (solve for $x$): $e^{\pi i x} -e^{-\pi ix} = 2yi$ Both $x$ and $y$ are real numbers, $y$ is given. I have no clue on how to solve it analytically. All I ...
2
votes
1answer
54 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
18 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
17 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
22 views

Euler's formula for off-center circle [on hold]

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
38 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
52 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 ...
4
votes
0answers
60 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 ...
13
votes
3answers
530 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
30 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
52 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
43 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 ...
0
votes
0answers
23 views

How many tiles are Symmetrical? [on hold]

We have a tape of type $1 * 2015$ had tile from tiles unit square in four different colors so as not exceed two tile of the same color (tile unit square, any tile from type $1*1$) How many tiles are ...
2
votes
3answers
108 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.
0
votes
1answer
49 views

Find all three numbers [on hold]

Find all I three numbers, that can be divided to $11$, and the result equal to sum of square every digit from digits that number I tried but, I couldn't complete.
4
votes
1answer
41 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 ...
2
votes
1answer
42 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 ...
6
votes
5answers
105 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
19 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; ...
1
vote
1answer
79 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
73 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.
1
vote
8answers
72 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
1answer
34 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
23 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 ...
2
votes
0answers
30 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
43 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
74 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
131 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: ...
7
votes
5answers
336 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 ...
0
votes
0answers
18 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 ...
-2
votes
0answers
27 views

get real and imaginary part of incomplete elliptic integral

I am trying to evaluate the integral: $$\int \frac{1}{\sqrt{-z^2+k_1z^3+k_2z^4+k_3}} \;\mathrm{d}z$$ where $k_1=0.133, k_2=0.15, k_3=2.746$. The answer has been evaluated in MAPLE to 10 decimal ...
1
vote
4answers
29 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
18 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 ...
8
votes
1answer
101 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 ...
2
votes
3answers
55 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 $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ ...
2
votes
1answer
51 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
37 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
86 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
163 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 ...
0
votes
1answer
21 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
92 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 ...
1
vote
1answer
25 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
32 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,\ ...
1
vote
1answer
44 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
40 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 ...