Complex Definition and 1000 Threads

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.

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  1. A

    Find The Impedance For Two Complex Impedances in Parallel

    Finding the series for the first part of the problem was easy, but for parallel, I'm not sure how to separate the real from the imaginary in the fractions after I add them together? So, I take: ##(1/(2+3i) + 1/(1-5i)^{-1}##, and after I combine the denominators and combine all terms, I end up...
  2. L

    Parallel RLC circuit complex impedance graphing

    ^^ as mentioned in the homework statement, the relevant equation is my worked out impedance for the circuit. I have attached a diagram of the circuit below.
  3. M

    I Trying to understand roots of quadratic equations

    I understand the basic maths but I am getting varying answers as to whether these are real distinct roots or not. Could you please explain the mechanism for deciding this. Thanks in anticipation.
  4. M

    Are these values of the following correct? (Complex cube root of unity)

    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...
  5. binbagsss

    I Question about partial derivative relations for complex numbers

    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...
  6. Euge

    POTW Cohomology Ring of Complement of Hyperplanes in Complex n-space

    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 $$H^*(\mathbb{C}^n \setminus \bigcup_{k = 1}^m \ker \lambda_k ; G)$$
  7. B

    Textbook 'The Physics of Waves': Derive Complex Solution Form for SHM

    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 -...
  8. F

    Insights Views On Complex Numbers

    Continue reading...
  9. M

    Fundamental matrix for complex linear DE system

    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...
  10. M

    Complex Eigenvalues of system of DE

    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!
  11. Tybalt

    A What causes a complex symmetric matrix to change from invertible to non-invertible?

    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...
  12. MatinSAR

    Why Is My Argument Calculation for Complex Number Incorrect?

    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...
  13. M

    I Question about branch of logarithms

    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...
  14. MatinSAR

    Please check my calculations with these complex numbers

    $$Re(e^{2iz}) = Re(\cos(2z)+i\sin(2z))=\cos(2z)$$$$e^{i^3} = e^{-i}$$ $$\ln (\sqrt 3 + i)^3=\ln(2)+i(\dfrac {\pi}{6}+2k\pi)$$ 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...
  15. pellis

    A How Does U(1) Double-Cover SO(2) for a Specified Angle?

    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...
  16. chwala

    Show that the given function is continuous

    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...
  17. Euge

    POTW Integration Over a Line in the Complex Plane

    For ##c > 0## and ##0 \le x \le 1##, find the complex integral $$\int_{c - \infty i}^{c + \infty i} \frac{x^s}{s}\, ds$$
  18. Euge

    POTW Solution to a Matrix Quadratic Equation

    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##).
  19. Z-10-46

    A Even with a whimsical mathematical usage, solutions are obtained!

    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?
  20. T

    I Continuity of Quotient of Complex Values

    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##...
  21. V

    Polarities of capacitor plates in a complex circuit

    (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...
  22. chwala

    Find in the form, ##x+iy## in the given complex number problem

    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.
  23. E

    I Numerical Solution of Complex Systems in GR

    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...
  24. mcastillo356

    B Is this a complex number at the second quadrant?

    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...
  25. chwala

    Solve the problem involving complex numbers

    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,
  26. C

    3x3 matrix with complex numbers

    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)...
  27. M

    Why Does Flipping the Denominator in Complex Fractions Give the Wrong Answer?

    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...
  28. VX10

    I A question about Young's inequality and complex numbers

    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...
  29. F

    Insights An Overview of Complex Differentiation and Integration

    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...
  30. chwala

    Find the GCD of the given complex numbers (Gaussian Integers)

    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...
  31. S

    I Noether currents for a complex scalar field and a Fermion field

    For a complex scalar field, the lagrangian density and the associated conserved current are given by: $$ \mathcal{L} = \partial^\mu \psi^\dagger \partial_\mu \psi -m^2 \psi^\dagger \psi $$ $$J^{\mu} = i \left[ (\partial^\mu \psi^\dagger ) \psi - (\partial^\mu \psi ) \psi^\dagger \right] $$...
  32. chwala

    Find the roots of the complex number ##(-1+i)^\frac {1}{3}##

    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...
  33. G

    Using complex numbers to solve for a current in this circuit

    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.
  34. O

    Symbolic integration of a Bessel function with a complex argument

    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...
  35. K

    How Can We Prove the Conjugate Transpose Property of Complex Matrices?

    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...
  36. A

    How to find z^n of a complex number

    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...
  37. person123

    I Newton-Raphson Method With Complex Numbers

    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...
  38. Euge

    POTW Limit of Complex Sums: Find $$\lim_{n\to \infty}$$

    Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
  39. U

    How to Calculate the Complex Op Amp in Circuit Diagram

    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...
  40. C

    Complex numbers problem |z| - iz = 1-2i

    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...
  41. C

    How Can I Solve a System of Equations With Complex Numbers?

    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...
  42. nomadreid

    I Applications of complex gamma (or beta) functions in physics?

    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...
  43. U

    Complex Integration Along Given Path

    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 +...
  44. T

    I Closed Form for Complex Gamma Function

    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...
  45. chwala

    Determine if the given set is Bounded- Complex Numbers

    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...
  46. chwala

    Solve ##z^2(1-z^2)=16## using Complex numbers

    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 ...
  47. R

    Understanding Complex Conjugates in QM (Griffiths pg. 13)

    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...
  48. J

    Linear operator in 2x2 complex vector space

    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...
  49. S

    B Phase Difference of Current & Voltage: Capacitors, Inductors & Complex Numbers

    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?
  50. H

    I Analysis of converting a DE into complex DE

    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...
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