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
97 views

Find the value of $x$

Find the value of $x$, $$\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$$ Help guys please, I have tried and I got, $x=-2, x=1$, and I think it's wrong
5
votes
4answers
67 views

Interesting summation question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to?

Question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to? $$\sum_{n=1}^{1729} \left[(-1)^n\cdot V(n)\right]$$ Where $$V(n)=a^n+b^n$$ My effort: I think I ...
0
votes
0answers
11 views

Application of Luca's theorem

Let, $p$ be a polynomial in $1$-complex variable. Suppose all zeros of $p$ are in the upper half plane $H=\{z\in \mathbb C|\Im(z)>0\}. $ Then , which are corrct ? $\Im\frac{p'(z)}{p(z)}>0$ for ...
1
vote
3answers
219 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
-2
votes
1answer
31 views

Consider the square defined by $0 \leq Re(z) \leq 1$ and $0 \leq Im(z) \leq1$. Determine … [on hold]

I have the following problem to solve: Consider the square defined by $0 \leq \operatorname{Re}(z) \leq 1$ and $0 \leq \operatorname{Im}(z) \leq1$. Determine the image of this square by the ...
8
votes
5answers
1k 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, ...
2
votes
3answers
54 views

Is there a simple way to define the $n$-th roots of the unity?

Is there a simple way to calculate the $n$-th roots of the unity? I gotta solve the equation $$\frac{z+1}{z-1}=\sqrt[n]{1}.$$
0
votes
2answers
30 views

Condition for the argument when complex numbers are written in polar form

In my text book it says that the complex number z(not equal to 0) can be written in polar form as $z = r(\cos\theta + i \sin\theta)$, where r = mod z greater than 0 is the modulus and $\theta = \arg ...
0
votes
0answers
61 views

How to calculate the Minimum of a set of Complex numbers? [duplicate]

Suppose you have 5 complex numbers $$2+4i,\ 6-3i,\ -9-7i,\ -12+23i,\ 3+4i.$$ How do you calculate the Minimum? And does it even make sense? If so, what would be a real world example? Thanks, Shane
0
votes
0answers
21 views

How to find a real function from a complex function.

I have the complex function $z\left(n\right) = i^{n} = \cos\left(\theta\left(n\right)\right) + i \sin\left(\theta\left(n\right)\right), \theta\left(n\right) = \frac{n \pi}{2},$ and I know that, on an ...
1
vote
1answer
22 views

Checking whether points form a polygon in complex plane

If z^8=(z-1)^8 then the roots are 1) concyclic 2) form a polygonal 3)none I found the roots to be 1+cot(k.pi/8) for k is a natural number and less than 8. Then couldn't figure it out.
0
votes
1answer
45 views

An analytic function such that $|f^2(z)-1|=|f(z)-1|\,|f(z)+1|<1.$

Let $f$ be an analytic function such that $$|f^2(z)-1|=|f(z)-1|\,|f(z)+1|<1.$$ on a non empty connected set $U$. Then (A) $f$ is constant. (B) $Im (f)>0$ on $U$. (C) $Re(f)\not = 0$ on $U$. ...
4
votes
3answers
71 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
36 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
57 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
77 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
40 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
146 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
98 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 ...
2
votes
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
71 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
58 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
46 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
55 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
112 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
60 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
417 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
59 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
98 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
64 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
39 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 [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: $R e^{i \theta}$. ...
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. ...