I have this problem in my book:
Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros.
In the picture below is the given solution for this:
I...
Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0.
I set up my normalization integral as follows:
A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1
After simplifying, and accounting for the fact that...
Hello everyone,
I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...
Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex?
For example, given
\begin{equation}
\begin{split}
\hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...
I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question.
I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...
Homework Statement
Write ##5-3i## in the polar form ##re^\left(i\theta\right)##.
Homework Equations
$$
|z|=\sqrt {a^2+b^2}
$$
The Attempt at a Solution
First I've found the absolute value of ##z##:
$$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$.
Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience...
Homework Statement
Find ##z## in ##z^{1+i}=4##. Is my solution correct
Homework Equations
##\log(z_1 z_2)=\log(z_1)+\log(z_2)## such that ##z_1, z_2\in \{z\in\Bbb{C} : (z=x+iy) \land (x\in\Bbb{R}) \land -\infty \lt y \lt +\infty\}##
##re^{i\theta}=r(\cos\theta + i\sin\theta)##
The Attempt at a...
Homework Statement
Refer given image.
Homework Equations
Expansion of determinant.
w^2+w+1=0 where w is cube root of 1.
The Attempt at a Solution
Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.
Hi.
If you have seen the above image which shows a parabola then you can also see that there is a colored portion of the parabola that have solution in "another dimension" - the "another dimension" can give me new numbers to form a solution of a function like f(x) = x2 + 1.
1. Is this "another...
Hi
I was hoping some of you would give me a clue on how to solve this complex number task.
z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2
I think there must be some nice looking way to solve it.
My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z|
After...
Hello everyone.
Iam reading about complex numbers at the moment ad Iam quite confused.
I know how to use them but Iam not getting a real understanding of what they actually are :-(
What exactly is the imaginary part of a complex number? I read that it could in example be phase....
Thanks in...
It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...
Homework Statement
I've used z* to mean z conjugate.
Given the equation z + 2iz* = 8 + 7i, express z in the form a + ib.
From SQA Advanced Higher Mathematics 2005 Exam Paper
Homework Equations
n/a
The Attempt at a Solution
I substituted a+ib and its conjugate in for z and z*, which, after...
Could you give me a hint how to attack this problem?
Find a complex number z = a+i*b such that f(t)=Re e^(z*t) where f(t)=cos(2*pi*t)
I have begun as follows:
e^((a+i*b)*t)=e^(a*t)*(cos(b)+i*sin(b))
Re e^(z*t)= e^(a*t)*cos(b)
What to do now?
Homework Statement
If Z1+Z2+Z3=0 and Z1*Z2 + Z2*Z3 + Z3*Z1=0 and Z1, Z2, Z3 are all complex, what is the value of
(|z1|+|z2|+|z3|)/(|z1*z2|+|z2*z3|+|z3*z1|)
Homework Equations
The Attempt at a Solution
I tried to multiply the equations by the product of all conjugates and reach some...
Homework Statement
So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve.
It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation:
Homework Equations...
I have been trying to show that
$$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$
Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$
I have tried by using the fact that
$$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C...
I'm having trouble figuring out to get the answers from the 2 equations. The phasors and complex numbers confuse me. Do I need to change the phasor form? How do I go about doing this thanks! (Not homework question im trying to figure this for my exam!)
Homework Statement
Express the complex number (−3 +4i)3 in the form a + bi
Homework Equations
z = r(cos(θ) + isin(θ))
The Attempt at a Solution
z = -3 + 4i
z3 = r3(cos(3θ) + isin(3θ))
r = sqrt ((-3)2 + 42)
= 5
θ = arcsin(4/5) = 0.9273
∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273))
a = -117
b...
Homework Statement
Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$.
The answers- 0, purely imaginary , purely real respectively.
Homework Equations
The Attempt at a Solution
I...
Homework Statement
Write this complex number in the form "a+bi"
a) cos(-pi/3) + i*sin(-pi/3)
b) 2√2(cos(-5pi/6)+i*sin(-5pi/6))
Homework Equations
my only problem is that im getting + instead of - on the cosinus side.(real number)
The Attempt at a Solution
a) pi/3 in the unit circle is 1/2 for...
The problem
The following equation ##z^4-2z^3+12z^2-14z+35=0## has a root with the real component = 1. What are the other solutions?
The attempt
This means that solutions are ##z = 1 \pm bi##and the factors are ##(z-(1-bi))(z-(1+bi)) ## and thus ## (z-(1-bi))(z-(1+bi)) =...
I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification.
It goes like this:
Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...
Homework Statement
[/B]
Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero.
The Attempt at a Solution
[/B]
(z-(1+i)(z-i) = Z^2-z-1-2iz+i
(Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i)
Z^2 term...
Homework Statement
How would I go about solving 1/z=(-4+4i)
The answer that I keep on getting is wrong
The Attempt at a Solution
[/B]
What I did
z=1/(-4+4i)x(-4-4i)/(-4-4i)
z=(-4-4i)/(16+16i-16i-16i^2)
z=(-4-4i)/32
z=-1/8-i/8
This is the answer that I got however it says that it is...
Homework Statement
in a given activity: solve for z in C the equation: z^3=1
Homework Equations
prove that the roots are 1, i, and i^2
The Attempt at a Solution
using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2)
it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution...
Homework Statement
Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle.
Homework Equations
The Attempt at a Solution
As far as I understand, I need...
Or basically anything that isn't a positive integer.
So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds.
If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)##
My proof below:
However I struggle to do this with...
Hello, I am enrolled in calculus 2. Just having started a section in our text book about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem:
∫ 1/(x^2+1)dx
I immediately...