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
4answers
89 views

Taking the square root of an imaginary number

We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota). The question is, ...
4
votes
3answers
68 views

The imaginary unit, $i,$ and an alternate representation.

Recently, I began working with both complex, and imaginary numbers, and I looked at the complex number $i^{n}.$ If $n = 0, i^{n} = 1,$ $n = 1, i^{n} = i = \sqrt{-1},$ $n = 2, i^{n} = i^{2} = i ...
0
votes
2answers
27 views

Complex Cube Roots - Argand Diagram Question

Suppose you have $3$ points on Argand diagram, evenly spread ($\frac{2\pi}{3}$ apart), represented by complex numbers $\alpha$, $\beta$ and $\gamma$, with moduli $\sqrt{2}$, if we take another complex ...
9
votes
6answers
6k views

How can you find the cubed roots of $i$?

I am trying to figure out what the three possibilities of $z$ are such that $$ z^3=i $$ but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you ...
1
vote
2answers
35 views

Why aren't these two properties of complex powers the same?

Let $z\in\mathbb{C}$ s.t. $z=u+iv$. As an example, take the square in this trivial manner: $(u+iv)^2=u^2-v^2+2iuv$. On the other hand taking the square using the properties of complex powers, i.e. ...
1
vote
1answer
50 views

Showing a function map to itself

Let $ D = \{ z \in \mathbb{C} : |z| < 1\}$. Fix $ w \in D$ and define $f: \bar{D} \to \mathbb{C}$ by $$f(z) = \frac{w-z}{1-\bar{w}z}$$ Show the following: $f$ maps $D$ to $D$ and $\partial D$ to ...
2
votes
2answers
71 views
+50

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 ...
2
votes
1answer
39 views

Roots of polynomial outside a vertical strip of $\mathbb C$

Let $P(z)$ be an arbitrary polynomial with real coefficients. I'd like to guarantee that all roots of $P$ have real parts outside the interval $(0, 1)$. Is there some simple condition on P that will ...
3
votes
4answers
141 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.
8
votes
1answer
93 views

For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
1
vote
1answer
34 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
1answer
68 views
0
votes
0answers
23 views

$(ab^{n+1})^{1/n}$ where $a,b \in \mathbb{C}$

Let $a,b \in \mathbb{C}$ and $n \in \mathbb{N}$. We can present $a$ and $b$ in polar form as \begin{equation} a = r_a \mathrm{e}^{i \theta_a} \quad \textrm{and} \quad b = r_b \mathrm{e}^{i \theta_b}, ...
3
votes
3answers
56 views

How do limits work in complex functions?

I don't quite understand one example in my notes it says. My query is this: I don't understand what the significance of $\theta$ is. Why does it matter that $\theta \in (-\pi,\pi]$? I see the ...
0
votes
0answers
16 views

The bilinear mapping that maps the given three points onto three given points

I need to solve this question: Find the bilinear mapping that maps the given three points onto the three given points in the respective order $$ 1,i,5\quad\text{into}\quad i,-2i,2. $$ The way I can ...
5
votes
2answers
57 views

System of equations in a,b,c,d

$a,b,c,d$ are complex numbers satisfying \begin{cases} a+b+c+d=3 \\ a^2+ b^2+ c^2+ d^2=5 \\ a^3+ b^3+ c^3+ d^3=3 \\ a^4+ b^4+ c^4+ d^4=9 \end{cases} Find the value of the following: ...
0
votes
0answers
44 views

Can the triangle function approximate the Gaussian curve for complex numbers?

I was thinking about approximating the Gaussian curve with a triangular curve. The graphs look like this: their respective functions are: $$ y_1(x) = t(x) = max(0, 1 - |x|)$$ $$ y_2(x) = e^{ - ...
-5
votes
3answers
56 views

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an abelian group [on hold]

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an Abelian group with the multiplication operation of complex numbers.
-3
votes
4answers
54 views

How do I prove :$z\bar{z}=i$ has no solutions in $\mathbb{R}$?

Is there someone who can prove me that: $z\bar{z}=i$ has no solution in $\mathbb{R}$, where $z$ is complex variable and $\bar{z}$ it conjugate ? Note: $i$ is imaginary unit. Thank you for any ...
1
vote
1answer
110 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
0
votes
1answer
59 views

Initial conditions for second order ODE with complex stiffness

I'm trying to find initial conditions to ensure systems of the form stay bounded $\ddot{x}_i+\sum_{j=1}^N k_{ij} x_j = 0, \quad k_{ij} \in \mathbb{C}$. For simplicity let's say the $k_{ij}$ lie in ...
4
votes
1answer
416 views

What are applications of Lagrange's identity?

I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = ...
-5
votes
3answers
62 views

What's the value of $i^i$? [duplicate]

What's the value of $i^i$?Is it real or imaginary?[$i$ here denotes imaginary number.]
0
votes
2answers
26 views

Triangle inequality with complex numbers.

Okay so I know that: $$|z|-|z_0| \leq |z-z_0|$$ and similarly that $$|z_0|-|z| \leq |z-z_0|$$ but in my book it states that since this is true then it is obviously true that $$||z|-|z_0||\leq ...
0
votes
2answers
57 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 ...
16
votes
2answers
1k views

Wild automorphisms of the complex numbers

I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could ...
4
votes
2answers
227 views

Simplifying an expression in $\Bbb{Q}(\zeta_p)$

Let $\zeta_p$ be a $p$-th root of unity, where $p$ is an odd prime number. I just came across the following expression: $$\frac{(\zeta_p^2-\zeta_p+1)^3}{\zeta_p^2(\zeta_p-1)^2}.$$ Can we simplify ...
1
vote
1answer
30 views

What is the interactive explanation of a number to the power $\sqrt{-1}$

What happens when a number is multiplied with itself i times, i.e a number $n \in \mathbb{C}$, what is the explanation of $n^i$ ? I have tried a few by myself:- $e^i = cos \; 1 + i sin\; 1$ and $i^i$ ...
0
votes
2answers
97 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
61 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
1answer
21 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
2answers
44 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? ...
2
votes
1answer
16 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
38 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 ...
1
vote
1answer
18 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
24 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: $R e^{i \theta}$. ...
3
votes
1answer
56 views

Solve $z^{1+i}=4$

$\def\Log{\operatorname{Log}}$ I have to solve $z^{1+i}=4$. Is there any easy way? I'm starting like this: $$e^{(1+i)\Log z}=e^{2\Log2}$$ Then I solve $$(1+i)\Log z=2\Log2$$ But I really doubt I ...
1
vote
1answer
111 views

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$.

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$. I have already proved that other functions are continuous by using that $f, g$ are continuous implies $f+g$ and $fg$ are continuous. ...
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
61 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
549 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?
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
0answers
273 views

The sum (or difference) of two irrational numbers

So far I that for any irrational number without a real part (that $-n=\overline{n}$) plus/minus any irrational number with the same restrictions equals another irrational number. However, I want to ...
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 ...
7
votes
2answers
132 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: ...
1
vote
1answer
45 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? [closed]

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 ...
0
votes
1answer
49 views

Find all three numbers [closed]

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
42 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 ...