Complex Definition and 1000 Threads

Homework Statement Consider a harmonic wave given by $$\Psi (x, t) = U(x, y, z) e^{-i \omega t}$$ where ##U(x, y, z)## is called the complex amplitude. Show that ##U## satisfies the Helmholtz equation: $$ (\nabla + k^2) U (x, y, z) = 0 $$ Homework Equations Everything important already in...

Homework Statement Prove that the function ## f(z)= 1/\sqrt{2}(\sqrt{\sqrt{x^{2}+y^{2}}+x}+i*sgn(y)\sqrt{\sqrt{x^{2}+y^{2}}-x})## is holomorphic on the domain ## \Omega = \left \{ z: z \neq 0, \left | \arg{z} \right | <\pi\right \} ## and further that in this domain ##f(z)^{2} = z. ##...

I have a problem to put the complex number in mode (1000/3, ∠36.87), apparently the division simbol gives some syntax error

Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...

I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and...

Homework Statement I have the following integral I wish to solve (preferably analytically): $$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$ where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...

Homework Statement Assume that the two balanced loads are supplied by an 840-V rms 60-Hz line. Load #1: Y-connected with 30+j40 Ω per phase, Load #2: balanced three-phase motor drawing 48 kW at a power factor of 0.8 lagging. Assuming abc sequence, calculate the complex power absorbed by the...

Homework Statement I have an integral $$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$ where j and c are complex, which I'd like to write in terms of ## \text{erf}## I'd like to know what would happen to the integral limits as I make the change of variables ##t = jp - c##. 1) As ##p## tends...

1. The problem statement, all variables and given/known Does the shape or profile of a moment arm impact the torque created at the axel or fulcrum point Homework Equations T=fd[/B]The Attempt at a Solution Please see sketch is the torque created at position 1 in position to correct?

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)...

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)?

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...

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.

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...

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.

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

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?

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 +...

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...

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...

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...

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...

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...

<<<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...

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

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!

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...

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 $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...

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...

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...

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

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...

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...

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...

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?

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...

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...

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}...

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 $p^2$.

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...

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...

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 =...

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...

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...

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 $t\to{\infty}$...

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!

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...

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...

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...

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...