What is Complex: Definition and 1000 Discussions

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.

View More On Wikipedia.org
Recent contents Top users
  1. SemM

    A Understanding Complex Operators: Rules, Boundedness, and Positivity

    Hi, from the books I have, it appears that some rules for operators, boundedness, positivity and possibly the definition of the spectrum regard real operators, and not complex operators. From the complex operator ##i\hbar d^3/dx^3 ## it appears that it can be defined as not bounded (unbounded)...
  2. Zaya Bell

    I Complex Numbers in Wave Function: QM Explained

    I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?
  3. E

    Engineering Part C: Solving Resistance in a Complex Circuit

    Homework Statement Part C of the following problem http://www.chegg.com/homework-help/questions-and-answers/problem-1-find-equivalent-resistance-rab-circuits-figp36-q8156743 2. Homework Equations The Attempt at a Solution Hi, to solve this problem, first i tried to ignore the two resistors...
  4. D

    MHB Prove the Following is True About the Complex Function f(z) = e^1/z

    Consider the function $f(z) = e^{1/z}$, Show that for any complex number ${w}_{0} \ne 0$ and any δ > 0, there exists ${z}_{0} ∈ C$ such that $ 0 < |{z}_{0}| < δ$ and $f({z}_{0}) = {w}_{0}$ I really don't know where to begin on this.
  5. N

    I How can I amplify and shift the phases of a complex waveform in MATLAB?

    I have a complex waveform in MATLAB that is of the form y = A1 ei * 2 * π * f * t + Φ I need to amplify each sample point of the waveform to an amplitude A2 and also for it to shift phase by φ. I therefore construct a complex waveform for amplification y = (A2 / A1 ) ei+φ Then with...
  6. J

    Residue at poles of complex function

    Homework Statement Homework Equations First find poles and then use residue theorem. The Attempt at a Solution Book answer is A. But there's no way I'm getting A. The 81 in numerator doesn't cancel off.
  7. M

    Problems in Classic Mechanics -- looking for complex problems to work on

    Can someone please share complex problems interrelated between Kinamatics, Force, Linear Momentum, Work and Energy along with final answers to cross-check.
  8. D

    MHB How Do You Calculate the Real and Imaginary Parts of \( e^{e^z} \)?

    Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f). I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?
  9. chwala

    Understanding Complex Number Equations: An Exploration

    Homework Statement if ## x + iy## = ## \frac a {b+ cos ∅ + i sin ∅} ## then show that ##(b^2-1)(x^2+y^2)+a^2 = 2abx##Homework EquationsThe Attempt at a Solution i let ## ... ##x + iy = ## \ a(b+cos ∅ - i sin ∅)##/ ##(b + cos ∅)^2 + sin^2∅ ##...got stuck here... alternatively i let ## b +...
  10. Drakkith

    Derivative of a Complex Function

    Homework Statement Find the derivative of ##f(z)=\frac{1.5z+3i}{7.5iz-15}## Homework EquationsThe Attempt at a Solution I had no difficulty using the standard derivative formulas to find the derivative of this function, but the actual result, that the derivative is zero, is confusing. For real...
  11. Drakkith

    I Is there a typo in the formula for dividing complex numbers?

    Quick question. While going over complex numbers in my book, I think I came across a typo and I wanted to be sure I had the right information. In the paragraph going over dividing complex numbers, my book has: ##|\frac{z_1}{z_2}|=|\frac{z_1}{z_2}|## That's obviously true. Should that be...
  12. A

    Torque with "complex" object (static/equilibrium)

    I have a general understanding of how torque works, at least for "simple" objects that can be drawn as a single "bar" under the effect of various forces. In this problem there is a slightly more "complex" object though, and I'd like to know if there is a way to solve it without doing what I did...
  13. binbagsss

    Complex scalar field, conserved current, expanding functional

    Homework Statement [/B] Hi I am looking at this action: Under the transformation ## \phi \to \phi e^{i \epsilon} ## Homework Equations [/B] So a conserved current is found by, promoting the parameter describing the transformation- ##\epsilon## say- to depend on ##x## since we know that...
  14. B

    A Laser beam represented with complex conjugate?

    Boyd - Nonlinear Optics page 5, there says 'Here a laser beam whose electric field strength is represented as $$\widetilde{E}(t) = Ee^{-iwt} + c.c$$But why is it written like this? Is it because the strength is the real part of the complex electric field? Then why doesn't he divide it by 2 after...
  15. J

    How to calculate the moment of inertia for complex 3D shapes

    <<<moved from another sub forum, no template>>Hi, I need to calculate the moment of inertia for the component in the attached image so that i can calculate the angular momentum. Is it possible? Overall i am trying to calculate the forces on this lug as it passes around a 3" radius at 2M a...
  16. A

    I Complex Fourier Series: Even/Odd Half Range Expansion

    Does the complex form of Fourier series assume even or odd half range expansion?
  17. Spinnor

    B String theory, Calabi–Yau manifolds, complex dimensions

    So in string theory at each point of Minkowski spacetime we might have a 3 dimensional compact complex Calabi–Yau manifold? We can have curved compact spaces without complex numbers I assume, what is interesting or special about complex compact spaces? Thanks!
  18. J

    Cauchy Integral of Complex Function

    Homework Statement Homework Equations Using Cauchy Integration Formula If function is analytic throughout the contour, then integraton = 0. If function is not analytic at point 'a' inside contour, then integration is 2*3.14*i* fn(a) divide by n! f(a) is numerator. The Attempt at a Solution...
  19. I

    A Second derivative of a complex matrix

    Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...
  20. JTC

    I Using Complex Numbers to find the solutions (simple Q.)

    Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function. We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time. Then, once done, if the forcing function was cosine, we choose the real part as the...
  21. T

    How do you always put a complex function into polar form?

    Homework Statement It's not a homework problem itself, but rather a general method that I imagine is similar to homework. For a given elementary complex function in the form of the product, sum or quotient of polynomials, there are conventional methods for converting them to polar form. The...
  22. B

    Complex Frequency Derivation-Magically Appearing "j"s

    Homework Statement From Hayt "Engineering Circuit Analysis". I'm just wondering how the imaginary "j" multipliers appeared. Homework EquationsThe Attempt at a Solution
  23. Ventrella

    B Complex products: perpendicular vectors and rotation effects

    My question is perhaps as much about the philosophy of math as it is about the specific tools of math: is perpendicularity and rotation integral and fundamental to the concept of multiplication - in all number spaces? As I understand it, the product of complex numbers x = (a, ib) and y = (c...
  24. sams

    I Finding Real and Imaginary Parts of the complex wave number

    In Griffiths fourth edition, page 413, section 9.4.1. Electromagnetic Waves in Conductors, the complex wave number is given according to equation (9.124). Calculating the real and imaginary parts of the complex wave number as in equation (9.125) lead to equations (9.126). I have done the...
  25. F

    Simplification of a complex exponential

    Homework Statement Is there a way to simplify the following expression? ##[cos(\frac {n \pi} 2) - j sin(\frac {n \pi} 2)] + [cos(\frac {3n \pi} 2) - j sin(\frac {3n \pi} 2)]## Homework Equations ##e^{jx} = cos(x) + j sin(x)## The Attempt at a Solution ##cos(\frac {n \pi} 2)## and...
  26. Thejas15101998

    I Operation on complex conjugate

    Why do we sandwich operators in quantum mechanics in such a way that the operator acts on the wavefunction and not on its complex conjugate?
  27. T

    MHB Additional solution for polar form of complex number

    Hi, I had a question I was working on a while back, and whilst I got the correct answer for it, I was told that there was a second solution to it that I missed. Here is the question. ] I worked my answer out to be sqrt(2)(cos(75)+i(sin(75))), however, it appears there is a second solution...
  28. S

    A Can imaginary position operators explain real eigenvalues in quantum mechanics?

    Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral : \begin{equation} \langle Bx, x\rangle \end{equation} when replaced by:\begin{equation} \langle Bix...
  29. S

    Proving Complex Number Equality

    Homework Statement ##z## is a complex number such that ##z = \frac{a+bi}{a-bi}##, where ##a## and ##b## are real numbers. Prove that ##\frac{z^2+1}{2z} = \frac{a^2-b^2}{a^2+b^2}##. Homework EquationsThe Attempt at a Solution I calculated \begin{equation*} \begin{split} z = \frac{a+bi}{a-bi}...
  30. lfdahl

    MHB Finding the Product of Distinct Roots: A Complex Challenge

    Let $r_1,r_2, …,r_7$ be the distinct roots (one real and six complex) of the equation $x^7-7= 0$. Let \[p = (r_1+r_2)(r_1+r_3)…(r_1+r_7)(r_2+r_3)(r_2+r_4)…(r_2+r_7)…(r_6+r_7) = \prod_{1\leq i<j\leq 7}(r_i+r_j).\] Evaluate $p^2$.
  31. K

    I String theory on complex spacetime, twistor string

    bosonic string theory requires 26 dimensions superstring theory requires 10, 9 spatial 1 dimension of time Witten has researched twistor string theory has there been any serious research with (super) string theory written on 4 complex -valued dimensions of spacetime? the additional dimensions...
  32. S

    I Types of complex matrices, why only 3?

    Hi, the three main types of complex matrices are: 1. Hermitian, with only real eigenvalues 2. Skew-Hermitian , with only imaginary eigenvalues 3. Unitary, with only complex conjugates. Shouldn't there be a fourth type: 4. Non-unitary-non-hermitian, with one imaginary value (i.e. 3i) and a...
  33. M

    I Interpretation of complex wave number

    Dear forum members, I'm wondering about the physical meaning of the imaginary part of a complex wave number (e.g., the context of fluid dynamics or acoustics). It is obvious that w = \hat{w} \mathrm{e}^{i k_z z} describes an undamped wave if k_z = \Re(k_z) and an evanescent wave if k_z =...
  34. K

    I Can complex spacetime solve the Higgs hierarchy problem?

    thus far the LHC hasn't found any evidence of SUSY or technicolor. thus far it's just 1 fundamental scalar there is an extensive literature on the Higgs hierarchy problem with various proposals and solutions offered has there been any scientific papers and research on the physical properties...
  35. J

    MHB Complex Variables - Max Modulus Inequality

    Suppose that f is analytic on the disc $\vert{z}\vert<1$ and satisfies $\vert{f(z)}\vert\le{M}$ if $\vert{z}\vert<1$. If $f(\alpha)=0$ for some $\alpha, \vert{\alpha}\vert<1$. Show that, $$\vert{f(z)}\vert\le{M\vert{\frac{z-\alpha}{1-\overline{\alpha}z}}\vert}$$ What I have: Let...
  36. J

    MHB Complex Variables - Solution of a System

    Suppose the polynomial p has all its zeros in the closed half-plane $Re w\le0$, and any zeros that lie on the imaginary axis are of order one. $$p(z)=det(zI-A),$$ where I is the n x n identity matrix. Show that any solution of the system $$\dot{x}=Ax+b$$ remains bounded as $t\to{\infty}$...
  37. S

    I Convert complex ODE to matrix form

    Hi, I have the following complex ODE: aY'' + ibY' = 0 and thought that it could be written as: [a, ib; -1, 1] Then the determinant of this matrix would give the form a + ib = 0 Is this correct and logically sound? Thanks!
  38. Jamz

    I Map from space spanned by 2 complex conjugate vars to R^2

    Hello, I would like your help understanding how to map a region of the space \mathbb{C}^2 spanned by two complex conjugate variables to the real plane \mathbb{R}^2 . Specifically, let us think that we have two complex conugate variables z and \bar{ z} and we define a triangle in the...
  39. J

    MHB Complex Variables - Legendre Polynomial

    We define the Legendre polynomial $P_n$ by $$P_n (z)=\frac{1}{2^nn!}\frac{d^n}{dz^n}(z^2-1)^n$$ Let $\omega$ be a smooth simple closed curve around z. Show that $$P_n (z)=\frac{1}{2i\pi}\frac{1}{2^n}\int_\omega\frac{(w^2-1)^n}{(w-z)^{n+1}}dw$$ What I have: We know $(w^2-1)^n$ is analytic on...
  40. J

    MHB Complex Variables - Zeros of Analytic Functions

    Studying for my complex analysis final. I think this should be a simple question but wanted some clarification. "Extend the formula $$\frac{1}{2i\pi} \int_\omega \frac{h'(z)}{h(z)}\, dz = \sum_{j=1}^N n_j - \sum_{k=1}^M m_k$$ to prove the following. Let $g$ be analytic on a domain...
  41. J

    I What does complex conjugate of a derivate mean?

    An exercise asks me to determine whether the following operator is Hermitian: {\left( {\frac{d}{{dx}}} \right)^ * }. I don't even know what that expression means. a) Differentiate with respect to x, then take the complex conjugate of the result? b) Take the complex conjugate, then...
  42. S

    Understanding Complex Plane and Finding Arguments: A Scientist's Perspective

    Homework Statement The picture below. Homework Equations cos2x=1-2sinx sin2x= 2sinxcosxThe Attempt at a Solution I got the modulus by using the Pythagoras theorem which is 2sin theta But I faced difficulty to find the argument. I have no idea why i end up with tan a (alpha) = cot theta which...
  43. S

    I Can a Hermitian matrix have complex eigenvalues?

    Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...
  44. R

    Simplification of complex expression

    Homework Statement For the expression: $$E=\frac{E_{0}}{2}\left(\exp\left[\frac{j\pi V}{2V_{\pi}}\right]+j\exp\left[-\frac{j\pi V}{2V_{\pi}}\right]\right)$$ I want to show that if ##V=m(t)-\frac{V_{\pi}}{2}##, then ##|E|^2## can be written as...
  45. S

    I How can I analyse and classify a matrix?

    Hi, I have a matrix of an ODE which yields complex eigenvalues and eigenvectors. It is therefore not Hermitian. How can I further analyse the properties of the matrix in a Hilbert space? The idea is to reveal the properties of stability and instability of the matrix. D_2 and D_1 are the second...
  46. Drakkith

    Using NVA to Solve for i in Polar Form | Complex Numbers Homework

    Homework Statement Use NVA to solve for ##i##. Enter your answer in polar form with the angle in degrees. Homework EquationsThe Attempt at a Solution My nodes are as follows: ##V_1## on the left middle junction, ##V_2## is the junction in the very center, and ##V_3## is the junction on the...
  47. X

    A Complex components of stress-energy tensor

    Hi All, I am evaluating the components of the stress-energy tensor for a (Klein-Gordon) complex scalar field. The ultimate aim is to use these in evolving the scalar field using the Klein-Gordon equations, coupled to Einstein's equations for evolving the geometric part. The tensor is given by...
  48. Milsomonk

    EOM for a complex scalar field

    Homework Statement Find the equations of motion for the Lagrangian below: $$ L=\partial_\mu \phi^* \partial^\mu \phi - V( \phi,\phi^* ) $$ Where : $$ V( \phi,\phi^* )= m^2 \phi^* \phi + \lambda (\phi^* \phi)^2 $$Homework Equations Euler Lagrange equation: $$ \partial_\mu \dfrac {\partial L}...
  49. D

    Peskin complex scalar field current

    Homework Statement i'm trying to calculate the charge operator for a complex scalar field. I've got the overal problem right but I'm confused about this: On page 18 of Peskin, it is written that the conserved current of a complex scalar field, associated with the transformation ##\phi...
  50. Math Amateur

    MHB Complex Integration - Conway - First Example on page 63 .... Section 1, Ch. IV .... ....

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding the first example on page 63 ... ... The the first...
Back
Top