2
votes
1answer
44 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
1
vote
3answers
53 views

Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
1
vote
1answer
24 views

To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
1
vote
2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
0
votes
4answers
68 views

Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
3
votes
1answer
70 views

Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
0
votes
2answers
27 views

Express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$ using Euler's identity

Using Euler's identity ($e^{in\theta}=\cos n\theta+i \sin n\theta$), express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$. Any ideas?
2
votes
1answer
61 views

Singularities of complex functions.

How do I determine the singularities of a function? What is a singularity? In the functions below which are the singularities? a)$$f(z)=\frac{1}{(z^4+2z)}$$ b)$$f(z)={e^{1/z}}$$
1
vote
3answers
38 views

Determining Laurent Series expansion and residues

Determining Laurent Series expansion and residues of $f(z)=\frac{z}{(z+1)(z+2)}$ around $z = -2$. What is the validity of the expanded region? What is $res(f, -2)$??
1
vote
4answers
62 views

Which way will produce the following integral?

Which way $\gamma$ will produce the following integral? $$\int\limits_{\gamma}\frac{3+i}{z^5 - z}dz = 0$$
0
votes
3answers
40 views

Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
0
votes
3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
1
vote
0answers
22 views

Curves composition with holomorphic function

Statement $(i)$ Let $\gamma:\mathbb R \to \mathbb C$ a $C^1$ curve. Let $v={\gamma}'(t_0)$ the complex number that one obtains from translating to the origin the tangent vector to $\gamma$ at ...
0
votes
1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
1
vote
2answers
61 views

Solve for x if $z$ is a complex number such that $z^2+z+1=0$

I was given a task to solve this equation for $x$: $$\frac{x-1}{x+1}=z\frac{1+i}{1-i}$$ for a complex number $z$ such that $z^2+z+1=0$. Solving this for $x$ is trivial but simplifying solution ...
0
votes
4answers
65 views

complex numbers quadratic equation question

how to solve $z^2 +3|z| = 0 , z$ complex ? treating the complex number as $a+bi $ or anything similar didnt help much...also solving like simple algebric equations also didnt prove effective and ...
4
votes
4answers
56 views

Proof of trigonometric identity $\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$

I was given a task of proving the following identity: $$\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$$ I am not looking for a solution, just some kind of a hint to start ...
2
votes
2answers
53 views

Inequality in complex numbers

Prove that for all $z\in \mathbb{C}$ $$\frac{\Vert z+i\Vert z\Vert \Vert}{\Vert z+1\Vert}\leq \frac{2\Vert z \Vert}{\Vert z \Vert +1}$$
0
votes
0answers
34 views

Basic complex geometry: Reflexion by a line. Where does $ \overline{z}$ go?

Where does $ \overline{z}$ go in the end? Shouldn't the formula be $f(z)= w^2 \overline{z} +2isw$? Thanks in advance. Original link: ...
-1
votes
2answers
55 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
0
votes
2answers
51 views

What is the polar form of -6i?

The module of -6i is 6 (the square root of 36), but $ tan\theta = -6/0$, meaning that the polar form $ 6(cos\theta + isen\theta) $ is also indefinite?
0
votes
1answer
20 views

Complex polynomial inequality

Let $P(z) = 2z - z^3 + 6z^4 + z^6$ with roots $\alpha_1, \alpha_2, \cdots, \alpha_6$. Using the identity $ ||z|-|\omega|| \leq |z+\omega| \leq |z| + |\omega|$ show that if $|z| > 3$, then $|2z - ...
0
votes
2answers
69 views

Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
0
votes
1answer
35 views

Under what conditions on $a,b$ is $1/(a+bi)=(1/a)+(i/b)$?

Question in proofs review in the complex numbers unit. I expressed $1/(a+bi) = (a-bi)/(a^2+b^2)$ I then separated the two terms in the denominator to get $a/(a^2+b^2)-bi/(a^2+b^2)$ I then equated ...
0
votes
1answer
22 views

Find the number of distinct elements.

Let $\omega$ denote a non-real cube root of unity. Then find the number of distinct elements in the set $\{ (1+\omega + \omega^2 + \cdots + \omega^n)^m | m,n \in \Bbb Z_+ \}$
1
vote
1answer
51 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
0
votes
1answer
34 views

Complex numbers exercise - homework

We have to prove that $z_1^{24n}+z_2^{24n}=2^{12n+1}$ if we know that a)$z_1z_2=2$ b)$z_1^3+z_2^3=-4$ I have tried many things but nothing worked so far
1
vote
5answers
52 views

Explain why there are two complex numbers z such that $|z| = 1$ and that satisfy the equation $|z| = |z-1|.$

I must find both such complex solutions and express them in Euler form and usual form. So it's been a while since I've touched the imaginary/real plane. However, from what I remember, $z = a + bi$. ...
-3
votes
1answer
78 views

Real and Imaginary [duplicate]

$$Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 2$$ $$Im\Big(({\frac{1+i}{1-i})^5\Big)} = 1$$ I got that $Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 1 \ne 2$ And, that ...
2
votes
2answers
99 views

Real and Imaginary

$$Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 2$$ $$Im\Big(({\frac{1+i}{1-i})^5\Big)} = 1$$ I got that $Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 1 \ne 2$ And, that ...
2
votes
1answer
72 views

Prove that $\sum_{n=0}^{\infty}e^{in\theta}$ is bounded

For my homework class, we need to prove that a certain series converges, for which it is useful to use that this series is bounded ($\theta \in (0,2\pi)$): $$\sum_{n=0}^{\infty}e^{in\theta}$$ I ...
0
votes
1answer
19 views

Is $\|x_1\|^2 + 2\|x_2\|^2 > - 2\Re(ix_1\overline{x_2})$ for complex numbers $x_1,x_2$

This is the last piece I need for a proof for a homework problem. Could someone explain whether or not this inequality must hold?
0
votes
4answers
59 views

Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
2
votes
1answer
66 views

How to show $\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$

How to show $$\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$$ I begin with $$\sin(x+iy) = \frac{e^{x+iy}-e^{-x-iy}}{2i} = \frac{e^xe^{iy}-e^{-x}e^{-iy}}{2i}$$ $$ = ...
1
vote
3answers
30 views

Finding an expression for the complex number Z^-1

So I want to find out an expression to express: $$z^{-1}$$ I know the answer is: $$z^{-1} = \frac{x-iy}{x^2+y^2}$$ But how would I go about proving this/the steps to this?
1
vote
1answer
25 views

A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
1
vote
1answer
27 views

Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
0
votes
2answers
52 views

Complex number polar form equation

I've been struggling with a complex numbers algebra question for a few days now, and the tutor says I still haven't got it right. Express $z_4 =−\sqrt{3} + i$ in polar form. Hence solve the ...
5
votes
1answer
194 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
1
vote
2answers
51 views

Determining all complex Z in the equation

Let $n \in \mathbb N$. Determine all complex numbers $z \in \mathbb C $ such that $ |z| ^{n-2} = 1.$ How would i begin this question, thanks!
1
vote
1answer
59 views

Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
0
votes
1answer
32 views

Complex numer equation

Let $n\in\mathbb{N}$. Determine all complex numbers $z\in\mathbb{C}$ such that $z^{n-1}$ = $\bar{z}$ . I'm not sure if I'm doing this question right, but would the solutions be $+ 1,-1$ or $0$?
0
votes
2answers
96 views

Complex number proof

Let f(x), g(x) $\in \mathbb C[x].$ Prove that if f(x) | g(x) and g(x) | f(x), then there exists a nonzero $c \in \mathbb C$ such that $f(x) = c * g(x)$ (You may use the fact that for any p(x), q(x) ...
2
votes
1answer
123 views

Remainders with complex numbers

Let $ f(x) \in C [x] .$ Suppose $ f(-1+i) = 2+5i $ and $ f(-2-i)=-3. $ Determine the remainder of f(x) divided by $(x+1-i)(x+2+i). $ How would i begin with this question, like how would i ...
1
vote
2answers
180 views

Determine all complex numbers z in equation:

Let $n\in\mathbb{N}$. Determine all complex numbers $z\in\mathbb{C}$ such that $z^{n-1}$ = $\bar{z}$ How would I begin this? Would I begin by saying $z=a+ib$ and expand and stuff?
1
vote
2answers
51 views

Finding complex Fourier coefficients

This is probably an easy question, but I'm a little bit stuck, so any help will be appreciated. PROBLEM Find the complex Fourier coefficients of: $$f(t) = \sin(2\pi t)$$ and $$f(t) = |\sin(2\pi ...
1
vote
2answers
83 views

Why $A\sin(2\pi ft) =\frac{A}{2j}(e^{j2\pi ft}-e^{-j2\pi ft})$ but not $\frac{A}{2}(e^{j2\pi ft}-e^{-j2\pi ft})$?

$$v_{\mathrm{in}}(t)=A\sin(2\pi ft) =\frac{A}{2j}\left(e^{j2\pi ft}-e^{-j2\pi ft}\right) \\ |H(f)|=|H(-f)|;\angle H(f) = -\angle H(f) \\ v_{\mathrm{out}}=H(f)v_{\mathrm{in}}=\frac{A}{2j}H(f)e^{j2\pi ...
0
votes
1answer
62 views

Example of holomorphic function from unit disc to itself

let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
0
votes
0answers
13 views

Calculating the DFT of a sequence following a mathematical expression

This is homework, so please don't give a full solution. Give a formula for $F_k$ for all $k$ where $f_n=4^n$ for all $n=0,\dots ,N-1$. I ended up abusing Wolfram Alpha and getting probably way ...