The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London (UCL). The Faculty, the UCL Faculty of Engineering Sciences and the UCL Faculty of the Built Envirornment (The Bartlett) together form the UCL School of the Built Environment, Engineering and Mathematical and Physical Sciences.
Given that x is a complex cube root of unity, we have ## x^{3}=1 ## but ## x\neq 1 ##, where the cube roots of unity are ## 1, \omega, \omega^2 ## and ## \omega, \omega^2 ## are the imaginary roots such that ## \omega^3=1, 1+\omega+\omega^2=0 ##.
Now we will consider three cases of the...
Apologies this is probably a very bad question but it's been a while since I have seen this.
I have ##z=x+iy##. I need to convert ##\frac{\partial \psi(z)}{\partial z}## , with ##\psi## some function of ##z##, in terms of ##x## and ##y##
I have ##dz=dx+idy##. so
##\frac{\partial \psi }{\partial...
Let ##n \ge 1## and ##1 \le m \le n##. Suppose ##\lambda_1,\ldots, \lambda_m : \mathbb{C}^n \to \mathbb{C}## are ##\mathbb{C}##-linearly independent linear functionals. For each abelian group ##G##, determine the cohomology ring $$H^*(\mathbb{C}^n \setminus \bigcup_{k = 1}^m \ker \lambda_k ; G)$$
TL;DR Summary: Question on deriving the complex irreducible solution form for simple harmonic motions based on time translation invariant.
Reference textbook “The Physics of Waves” in MIT website:
https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/
Chapter 1 -...
For this problem,
I am trying to find the fundamental matrix, however, the eigenvalues are both imaginary and so are the eigenvectors. That is, ##\lambda_1 = 4i, \lambda_2 = -4i##
##v_1 = (1 + 2i, 2)^T##
##v_2 = (1 - 2i, 2)^T##
So I think I just have an imaginary matrix? This is because the...
For this problem,
Can someone please explain to me how they got from the orange step to the yellow step?
I am confused how the two expressions are equivalent.
Thanks!
I'm trying to get an intuitive grasp of why an almost imperceptible change in the off-diagonal elements in a complex symmetric matrix causes it to change from being invertible to not being invertible. The diagonal elements are 1, and the sum of abs values of the off-diagonal elements in each row...
Question 1: Find the modulus and argument of ##z=-\sin \frac {\pi}{8}-i\cos \frac {\pi}{8}##.
The modulus is obviously 1. I can't prove that the argument is ##\frac {-5\pi} {8}##. I think ##\frac {-5\pi} {8}## is not correct ...
What I've done:
$$\tan \theta=\cot \frac {\pi}{8}$$$$\tan...
I've read a proof from Complex Made Simple (David C. Ullrich)
Proposition 4.3. Suppose that ##V## is an open subset of the plane. There exists a branch of the logarithm in ##V## if and only if there exists ##f \in H(V)## with ##f'(z) = \frac{1}{z}## for all ##z \in V##.
Proof: One direction is...
$$Re(e^{2iz}) = Re(\cos(2z)+i\sin(2z))=\cos(2z)$$$$e^{i^3} = e^{-i}$$ $$\ln (\sqrt 3 + i)^3=\ln(2)+i(\dfrac {\pi}{6}+2k\pi)$$
Can't I simplify these more? Are they correct?
Final one:## (1+3i)^{\frac 1 2}##
Can I write in in term of ##\sin x## and ##\cos x## then use ##(\cos x+i\sin x)^n=\cos...
I'm trying to find an explicit example showing exactly how the U(1) “circle group” of complex numbers double-covers 2D planar rotations R(θ) that form the rotation group SO(2).
There are various explanations available online, some of which are clear but seem to be at variance with other...
Refreshing... going through the literature i may need your indulgence or direction where required. ...of course i am still studying on the proofs of continuity...the limits and epsilons... in reference to continuity of functions...
From my reading, A complex valued function is continous if and...
Let ##A## be a complex nilpotent ##n\times n##-matrix. Show that there is a unique nilpotent solution to the quadratic equation ##X^2 + 2X = A## in ##M_n(\mathbb{C})##, and write the solution explicitly (that is, in terms of ##A##).
Hello everyone,
Here, we observe that the familiar properties of the real logarithm hold true for the complex logarithm in these examples.
So why does a whimsical mathematical use of real logarithm properties yield coherent solutions even in the case of complex logarithm?
Hey all,
I have a very simple question regarding the quotient of complex values. Consider the function:
$$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$
where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...
(a) I think the top plate of C5 could end up with either + or - charge, and not necessarily + charge as shown. This is because the connected plates of C1, C5 and C3 form an isolated system to which we can apply the law of conservation of charge i.e. Total charge just before transient currents...
This is the question as it appears on the pdf. copy;
##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]##
My approach;
##\dfrac{3π}{4}=135^0##
##\tan 135^0=-\tan 45^0=\dfrac{-\sqrt{2}}{\sqrt{2}}##
therefore,
##z=-\sqrt{2}+\sqrt{2}i##
There may be a better approach.
Please help me confirm that I understand this correctly.
Imagine a system comprised of two black holes orbiting each other, which will eventually merge. At any point in time we describe the stress-energy tensor of the system. Assume that we could solve the EFE's for every point (t,x,y,z). This...
Hi, PF, so long, I have a naive question: is ##\pi+\arctan{(2)}## a complex number at the second quadrant? To define a single-valued function, the principal argument of ##w## (denoted ##\mbox{Arg (w)}## is unique. This is because it is sometimes convenient to restric ##\theta=\arg{(w)}## to an...
Hello guys,
I am refreshing on complex numbers today; kindly see attached.
ok for part (a) this is a circle with centre ##(\sqrt{3}, -1)## with radius =##1## thus, we shall have,
The attempt at a solution:
I tried the normal method to find the determinant equal to 2j. I ended up with:
2j = -4yj -2xj -2j -x +y
then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options:
1) 0= y(-4j-j^2) -x(2j-1) -2j
2)...
My first method to simplify the fraction is to to I flip ##\frac{5}{3}## up I get ##2 \times \frac{3}{5} = \frac{6}{5}##
Method 2: if I flip 3 up I get ##\frac{2}{5} \times \frac{1}{3} = \frac{2}{15}##.
Method 3: I could use it multiply ##\frac{3}{3}## since this is the same as mutlipying by...
Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks.
Analysis is about differentiation. Hence, complex differentiation will be my starting point...
Hello guys,
I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean?
##z=1.2-1.4i## is rounded down to ##z=1-i##?
I can see from here they came up with simultaneous equation i.e
##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
For a complex scalar field, the lagrangian density and the associated conserved current are given by:
$$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$
$$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$...
Kindly see attached...I just want to understand why for the case; ##(-1+i)^\frac {1}{3}## they divided by ##3## when working out the angles...
Am assuming they used;
##(\cos x + i \sin x)^n = \cos nx + i \sin nx## and here, we require ##n## to be positive integers...unless I am not getting...
First I solved 4+j3, which I squared 4 and 3 to equal 16 and 9 then I added them to get 25 and then I got the square root of 25 = 5.
Then I plugged it back in to the equation.
[50/(5)(50)+100] x 150 to get 50/350x 150= 1/7(150)= 21.42. I've attached the correct answer.
Hello all
I am trying to solve the following integral with Mathematica and I'm having some issues with it.
where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by
Where delta is a coefficient.
Due to the complex arguments I'm integrating the...
TL;DR Summary: For every Complex matrix proove that: (Y^*) * X = complex conjugate of {(X^*) * Y}
Here (Y^*) and (X^*) is equal to complex conjugate of (Y^T) and complex conjugate of (X^T) where T presents transponse of matrix
I think we need to use (A*B)^T= (B^T) * (A^T) and
Can you help...
Hello! (Not sure if this is pre or post calc,if I am in the wrong forum feel free to move it)
So im given this complex number ## z = \frac{6}{1-i} ## and I am susposed to get it in polar form as well as z = a+bi
I did that; z = 3+3i and polar form ##z =\sqrt{18} *e^{\pi/4 i} ##
Now Im...
I'm trying to code Newton Raphson's method for finding zeros. I realize that even if the solution is real, it's possible for guesses to be complex. For example:
$$y=\sqrt{x-6}-2$$
While 10 is a valid real root, for any guess less than 6, the result is complex.
I tried to run the code allowing...
Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
TL;DR Summary: How to calculate the operational amplifiers in the circuit diagram
Hello Everyone,
I am trying to learn the circuit diagram of one of a device in which I will be doing modifications as a part of my Masters's Research to make it performance better. My background is in Mechanical...
Here is my attempt(photo below), but somehow the solution in the textbook is z= 2 - (3/2)i, and mine is z=(-3/2) +2i.
Can someone please tell me where I am making a mistake? I suppose it's something with x being part of the real part of the 1st complex number and x being part of an imaginary...
How can I solve a system of equations with complex numbers
2z+w=7i
zi+w=-1
I have tried substituting z with a+bi and I have tried substituting w=7i-2z but didn't get anything useful.
Edit: also, I've tried, multiplying lower eq. with -1 so that I can cancel w but I get stuck with 2z and zi and...
An example of physical applications for the gamma (or beta) function(s) is
http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf
(I refer to the beta function related to the gamma function, not the other functions with this name)
The applications in Wikipedia...
From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
Hey all,
I was wondering if there was an equivalent closed form expression for ##\Gamma(\frac{1}{2}+ib)## where ##b## is a real number.
I came across the following answer...
My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say...
The problem is as shown...all steps are pretty easy to follow. I need help on the highlighted part in red. How did they come to;
##z^4+8z^2+16-9z^2=0## or is it by manipulating ##-z^2= 8z^2-9z^2?## trial and error ...
Am looking at page 13 of QM by Griffiths - have become stuck on minor point. He is proving that a normalised solution of Schrodingers eqn stays normalised. The bit I don't get is how can you just take the complex conjugate of Schrodingers eqn and assume its true. (ie how does he get from Eqn...
Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2?
____________________________________________________________
An ordered basis for C2x2 is:
I don't...
how does capacitors and inductors cause phase difference between current and voltage? how does complex number come into play to explain the relation between phase of current and voltage?
In Lecture 7, Prof. Arthur Mattuck (MIT OCW 18.03) taught that the following equation
$$
y’ +ky = k \cos(\omega t)$$
can be solved by replacing cos(ωt) by ##e^{\omega t}## and, rewriting thus,
$$
\tilde{y’} + k\tilde{y}= ke^{i \omega t}
$$
Where ##\tilde{y} = y_1 + i y_2##. And the solution of...
Consider the equation of motion for a simple harmonic oscillator:
##m\ddot {x}(t)=-kx(t).##
The solutions are
##x(t)=Ae^{i\omega t}+Be^{-i\omega t},##
where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the...
Hello,
I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions.
As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...