Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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2
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1answer
25 views

Complex Numbers - Finding Limits

$$\lim_{z\to 1+i}\frac{z^4 + 2i}{iz-3}$$ Attempt: I substituted $z = 1+i$ in the numerator and denominator: Since $i^2 = -1$ I got $(1+i)^4 = -4$ So, $$\frac{-4 + 2i}{i-4}$$
0
votes
1answer
16 views

Find the Limits Of Complex Functions

$$\lim_{z\to \infty}\frac{3z^2 + 2z - i}{2iz^2 - 1}$$ Attempt: I replaced z with 1/z and solved it. I got this $$\lim_{z\to \infty}\frac {2i-z^2}{3+2z-iz^2}$$
-3
votes
1answer
47 views

All solutions to $ z^{4} = -4 - i16 \sqrt{5} $ [on hold]

I am working on some exercises for my introduction to complex variables class and I have no idea how to solve this question. Given that $ (\sqrt{5} - i)^{4} = z^{4} = -4 - i16 \sqrt{5} $ Find ...
0
votes
0answers
29 views

real vs complex numbers

Can someone write REAL numbers in rectangular form as well? And if so, is it useful? For example: On the complex plane, (x + yi) is x units on the Real x axis and y units on the Imaginary y axis. If ...
3
votes
3answers
58 views

How to show that $\sqrt[3]{-1+\sqrt{-7}}+\sqrt[3]{-1-\sqrt{-7}}$ is a real number at a time before the invention of complex numbers

I have read this PDF from ocw.mit.edu about complex numbers. There is one interesting question: Imagine yourself at the time, when complex numbers had to be invented yet. How to show that ...
0
votes
1answer
20 views

How do I write this complex number in exponential form?

$$ -4 - i 16\sqrt{5}$$ Example: I know we can write $-8-i8\sqrt{3}$ as $16e^{i(-2\pi/3 + 2k\pi)}$ where $k = 0,\pm1, \pm2,....$
1
vote
2answers
27 views

Points on a straight line (Complex Analysis)

I encouter a problem in complex analysis course : Let $a, b, $ and $c$ be three distinct points on a straight line with $b$ between $a$ and $c$. Show that $\frac{a-b}{c-b} \in \mathbb{R}_{<0}$. ...
0
votes
2answers
28 views

General Formula for Principle Square Root of Complex Number

How can I prove that $ \sqrt{z} = \sqrt{|z|} \frac{(z + |z|)}{|z+|z||} $ without using mathematical induction, and if I cannot -- how would I go about using induction in the set of complex numbers ?
-4
votes
1answer
33 views

Algebra,complex numbers home work problem [on hold]

Please I want the solution of this problem : $z= \dfrac{(2-i) \cdot (x+4i)}{3-4i}$ and $|z|=2$ then $X=?$
0
votes
1answer
21 views

Sum of unitary complex numbers

Let us define: $$\varphi(x,n,t):=\frac{1}{n}\sum_{y=1}^n \left| \sum_{k=1}^n e^{2ik\pi (x-y)/n + 2i \sin(2k\pi/n) t} \right|$$ Does somebody have an idea how to prove that $$ \sup_{x=1,...,n} ...
1
vote
1answer
20 views

A question about complex using geometric.

Let $z_{1}$, $z_{2}$, and $z_{3}$ be three distinct complex numbers. Prove that these numbers are collinear if and only if the quotient $(z_{3}-z_{1})$ \ $(z_{2}-z_{1})$ is a real number. I have been ...
0
votes
4answers
40 views

Prove that $az^n+b\overline{z}^n=0$ does not have any complex solutions except for $0$ [on hold]

Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$? The first one seems very obvious, but ...
3
votes
7answers
123 views

What does $\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3} $ imply?

I'm having trouble understanding what the following equality implies. $$\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3}.$$ I suspect that this means that the points form the vertices of an ...
2
votes
1answer
28 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
1
vote
2answers
44 views

Complex differentiability of $f(z)=|z|$

Why is the absolute value function $f : \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|$ not complex differentiable at any point $z_0$ in the plane?
0
votes
1answer
28 views

Complex variables Open ball [on hold]

Let $f(z) = \frac1z$ be inversion. Given a real number $a$, let $R_a = \{z \in C : Im(z) < a\}$. Why is $f(R_a)$ an open disk, provided $a < 0$. What happens when $a \ge 0$?
0
votes
2answers
35 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
0
votes
1answer
60 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $a_0 , a_1 , a_2 , b_0 , b_1 , b_2 \in \mathbb{C} $ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( 1 + j b_0 ) ( 1 + j^2 b_0 ) ...
3
votes
2answers
31 views

Square Rooting Back To Real Dimension

As we all know, square rooting -1 (a real number) opens up the "imaginary" dimension (defined by the presence of iota). We can return from the imaginary dimension back to the real dimension by ...
1
vote
3answers
36 views

Can every polynomial be factored into constant and linear complex factors?

That is, can any polynomial, $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1+a_0$, be expressed $b_0\left(x + b_1\right)\left(x + b_2\right)\ldots \left(x + b_n\right)$ where $b_i \in \mathbb{C}$?
-2
votes
2answers
79 views

What is wrong with my proof: $-1 = 1$? [duplicate]

I have some theories about why this could by wrong but I still haven't something that convinces me. What is wrong with this proof: $ -1 = i^2 = i.i = \sqrt{-1}.\sqrt{-1} = \sqrt{(-1).(-1)}= \sqrt1 = ...
2
votes
3answers
89 views

Confused with imaginary calculus

So $i$ is the complex unit and $n \in \mathbb{N} $. $$e^{2 \pi \ n \ i} = 1$$ $$1^{2 \pi \ n \ i} = 1$$ $$(e^{2 \pi \ n \ i})^{2 \pi \ n \ i} = e^{-4\pi^2 \ n^2}$$ $$e^{-4\pi^2 \ n^2} \neq 1$$ I’m ...
4
votes
0answers
37 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
0
votes
1answer
29 views

Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
2
votes
1answer
26 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
2
votes
1answer
22 views

Length of a complex vector

From the definition of inner product in $\mathbb{F}^n$ $$\textbf{a}\cdot\textbf{a}=\sum\limits_{k=1}^na_{k}\overline{a_{k}}$$ Say $a_{k}=x_{k}+iy_{k}$, then ...
0
votes
0answers
36 views

proof of a vector identity

In an exercise I am asked to prove the following vector identity: $$\textbf{a}\cdot\textbf{b}=\frac{1}{4}\big(|\textbf{a}+\textbf{b}|^{2}-|\textbf{a}-\textbf{b}|^{2}\big)$$ Both the dimension of the ...
-1
votes
1answer
59 views

Complex transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ [on hold]

Under the transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ is transformed to (a) $\{z\in \mathbb C:0<\operatorname{arg}(z)<\pi\}$ (b) $\{z\in \mathbb ...
0
votes
1answer
48 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
-3
votes
2answers
32 views

Modulus of a complex expression [on hold]

Show that for a complex number $a$, $$\left\lvert\frac{z-a}{1-\bar{a}z}\right\rvert = 1$$ for $|z| =1$ and $\bar{a}z ≠ 1$. I've tried to show that if $|z-a| = |1-\bar{a}z|$ then it's true but to no ...
8
votes
2answers
156 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
0
votes
1answer
40 views

stuck with a cubic equation

I picked an equation $0= x^3 +x^2 -2x -1$ I plotted it with geogebra, to see if it had more than $1$ real root. It definitely cuts the $x$-axis $3$ times. But when I checked wolfram alpha, to see ...
5
votes
0answers
54 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
2
votes
1answer
22 views

Seperating points in the complex plane

Given a finite set of points say $p_1,p_2, \ldots, p_n$ in the complex plane, how do I find another point $q$ such that ray $R_i$ joining $q$ to $p_i$ are all distinct. I would be happy with any kind ...
3
votes
0answers
82 views

Limits of sequences connected with real and complex exponential

Let us denote $S_{n}(x)=1+\frac{x}{1 !}+\frac{x^{2}}{2!}+ ... + \frac{x^{n}}{n!}$. How could be calculated the limit $$L(x)=\lim_{n\to \infty}\frac{S_{n}(n x)}{e^{n x}}=\lim_{n\to ...
0
votes
1answer
46 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
1
vote
0answers
40 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
-6
votes
1answer
67 views

you know root square of -1, what is the larger of the square? [closed]

there is a square ABDC, $BD = \sqrt{-1}$ what is the value of AB=BC=DC=AD?
7
votes
3answers
248 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
1
vote
2answers
35 views

Polar form equations on the unit circle

If $l \in [0, 2 π)$, $k, n \in N$, proof the following equations: $$\mid{e^{i k l/n} - e^{i (k-1) l/n}}\mid = \mid e^{i l/n} - 1\mid$$ and: $$\lim_{n \to \infty} \sum_{k = 1}^n \mid e^{i k l/n} - ...
1
vote
1answer
17 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
0
votes
0answers
19 views

Example on Complex Number using De moivrs theorem [closed]

Prove That ( √3+i)^14 + ( √3-i )^14 = 2^14
2
votes
1answer
42 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
-5
votes
1answer
47 views

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. [closed]

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. This is a question on complex numbers.
2
votes
0answers
62 views

Can an ordered field contain complex numbers?

I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it ...
21
votes
6answers
2k views

Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
2
votes
1answer
18 views

Sixth root of -64 using Euler's formula and De Moivre's theorem

I am attempting to solve: $$(-64)^{\frac{1}{6}}$$ Using the relation: $$a+bi=re^{i(\tan^{-1}(\frac{b}{a})+2\pi n)}$$ And then applying De Moivre's theorem: ...
3
votes
0answers
51 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
0
votes
1answer
63 views

Find roots of $ω^x+(ω^x)^2+1=x$ [closed]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
0
votes
1answer
53 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...