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

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1
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2answers
14 views

Proving an Inequality Involving the Modulus of the Difference of Moduli

Prove the following inequality and give necessary and sufficient conditions for equality. $\left| |z|-|w| \right| \leq |z-w|$ for complex numbers $z$ and $w$. I have the following: By definition ...
0
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3answers
35 views

Partial fractions of $1/(z^2+2)$

How does one split $1/(z^2+2)$ into partial fractions? Normally I would factorise, but I cannot spot the solutions of $z^2+2=0$.
0
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1answer
16 views

ML inequaility problem, finding M

I have attempted the following problem, is my method and logic correct? I am looking for a upper bound of; $f(z)=z^2$. My line is from $1-2i$ to $1+2i$. Since we have no change in $x$, length is ...
3
votes
1answer
64 views

Prove that $f(z)=z^2$ is continuous.

Prove that $f(z)=z^2$ is continuous for all complex and real values of $z$. What I've got so far is: Given $ \epsilon >0$ and $|z-z_0|<\delta$ after some calculations (which I've checked with ...
2
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1answer
67 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
0
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1answer
16 views

Factorisation of polynomials in C [on hold]

How can I factorize this term in $\mathbb{C}$? Any further explanation will be appreciated! $$z^2-3z+4$$
3
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1answer
48 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
0
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2answers
35 views

Is there any geometric interpretation or significance of the complex roots of a derivative?

I was doing some reading online when I stumbled here and learned about this geometric way of viewing the complex roots of a function. It got me thinking; the zeros of the derivative of a function $f$ ...
4
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4answers
793 views

Euler's formula, is this true? [duplicate]

Let me have a function such as $ f(k) = \exp( 2 \pi i k ) $, where $k$ is real value. $$ f(k) = \exp( 2 \pi ik ) = (e^{ 2 \pi i} )^{k} =1^{k} = 1.$$ I think if this is true, there is no need to ...
0
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1answer
17 views

Show that a complex map is onto

I consider $\mathbb{C}$ as a real vector space. For $(a,b) \in \mathbb{C}^{2}$, consider the map : $F_{a,b} \, ; \, \mathbb{C} \, \rightarrow \, \mathbb{C}^{\ast}$ such that : $$ \forall z \in ...
0
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0answers
26 views

Absolute square in deriving Fourier transform variance

I'm having some trouble understanding how to derive the variance of the Fourier transform. This is for an image, i.e., it's a 2D transform. The variance is $|\hat{I}(\xi,\eta)|^2$, the absolute ...
4
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0answers
75 views

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ I tried with polar coordinate. So let $z=r(\cos \theta +i \sin \theta)$. the LHS $=|2z|=2|z|=2r$ The RHS, I do some ...
1
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2answers
35 views

Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?

Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
1
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4answers
40 views

Simplification of Complex Number.

I would appreciate any hints for the following problem: Given that $z=\dfrac{1-\cos4\theta+i\sin4\theta}{\sin2\theta+2i\cos^2\theta}$ show that $\vert z\vert=2\sin\theta$ and arg $z=\theta$ ...
1
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2answers
42 views

Defining the equation of an ellipse in the complex plane

Usually the equation for an ellipse in the complex plane is defined as $\lvert z-a\rvert + \lvert z-b\rvert = c$ where $c>\lvert a-b\rvert$. If we start with a real ellipse, can we define it in ...
2
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1answer
28 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
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1answer
17 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
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1answer
50 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
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0answers
30 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 ...
4
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3answers
61 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
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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
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2answers
28 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
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2answers
29 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 ?
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1answer
37 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 $X=?$
1
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0answers
25 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
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1answer
22 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
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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
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7answers
124 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
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1answer
31 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
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2answers
45 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
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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 $a \ge 0$?
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2answers
36 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
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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
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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
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3answers
37 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 $b_i \in \mathbb{C}$?
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2answers
82 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
90 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
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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
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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
28 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
27 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
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0answers
40 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 ...
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1answer
60 views

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

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
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1answer
50 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
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2answers
32 views

Modulus of a complex expression [closed]

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
157 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
41 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
89 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 ...