Function Definition and 1000 Threads

Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.

Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}## The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...

The book's procedure for the "shooting method" The point of this program is to compute a wave function and to try and home in on the ground eigenvalue energy, which i should expect pi^2 / 8 = 1.2337... This is my program (written in python) import matplotlib.pyplot as plt import numpy as...

Suppose we have a piecewise function f(t) = exp(c*t) when 0 <= t < 2 and f(t) = 0 when t >= 2. Can the above be rewritten as f(t)= exp(at)*[H(t-0) - H(t-2)], H is a heaviside function.

The problem of my question is the b part below: I know that the potential energy is just the gravitational potential energy, which is mgh(𝜃) = mg[(R+b/2)cos𝜃 +R𝜃sin𝜃], derived from the geometry. The equilibrium point is at 𝜃=0 and the system is a stable equilibrium for R>b/2. However, I have no...

Hi, I have pasted two improper integrals. The text has evaluated these integrals and come up with answers. I wanted to know how these integrals have been evaluated and what is the process to do so. Integral 1 Now the 1st integral is again integrated Now the text accompanying the integration...

I'm given the following density of states $$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$ where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$...

Some questions: Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well. Since we have no complex components. I am guessing that the ##\psi *=\psi##. If...

$$H=-J\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1}$$ There is no external magnetic field, so the Hamiltonian is different than normal, and the spins $\sigma_i$ can be -1, 0, or 1. The boundary conditions are non-periodic (the chain just ends with the Nth spin) $$Z=e^{-\beta H}$$...

On the image you can see a photon starting at point A at t=0. The photons travels along the sine function and arrives point C. I knot that this takes T=λ/c. But this is the time for a object traveling directly from the origin to point C and not along the sine wave! If the photon travels...

If we have that quotient of heats ##Q_2/Q_1=f(t_2,t_1)##, where ##t_1,t_2## are emirical temperatures. Is this function satisfies : ##f(t_2,t_1)=f(t_2-t_1,0)## I try prove it with Taylor series of two variables, but i can't prove anything.

I've got the answer for (a). It's k = 0.78 N/m. I'm having problems with (b). I know that the equation of displacement in this case should either be : x(t) = Asin(ωt + φ) or x(t) = Acos(ωt - φ) where A = amplitudeFrom what I understand, both the equation above should give the same result as...

My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...

Ok Just have trouble getting this without a function..

I was thinking about a problem I had considered a long time ago in some thread, finding an example of a wave function ##\displaystyle \psi (x) =e^{iax}\phi (x)## with ##\displaystyle\phi (x)## being periodic with period ##\displaystyle L## and the corresponding Schrödinger equation...

I conducted a mass-sprig experiment to see how stiffness of a spring and mass affect the frequency of oscillation. In addition to this to this i have to plot a graph to show displacement,velocity and acceleration of the mass as a function of time.From my research online For the displacement as...

When I type in this: D [ Re[ Exp[u + 10*I] ], u ] /. u->0.5 I get this output: Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?

Let f : N −→ R and f(x) = √ 9x The domain is all natural numbers: {0, 1, 2, 3, ...} The codomain is all real numbers. The range i believe is [0, +infinity) I believe that although the above is a function since every input of x provides a output that fits in our codomain. I also believe that...

The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$ Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z)) $$ The recurrence relations for the former are the same as those of the first kind, for which one of the relations is: $$...

Let's say you have a tensor u with the following components: $$u_{ij}=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'$$ Where G is a Green function, and g is just a normal well behaved function. My question is what is the square of this component? is it...

##f(x) = 6^x + 3^x + 6^{-x} + 3^{-x} + 2## But, ## AM >= GM## So, ##f(x) >= 5 * 2 ^ {\frac{1}{5}}## But this is not the case. According to the graph, it is ## f(x) >= 6##. If I do the same thing without considering the constant '2' then I am getting the answer. let ##g(x) = 6^x + 3^x +...

My attempt: According to the implicit function theorem as long as the determinant of the jacobian given by ∂(F,G)/∂(y,z) is not equal to 0, the parametrization is possible. ∂(F,G)/∂(y,z)=4yzMeaning that all points where z and y are not equal to 0 are possible parametrizations. My friend's...

Through some calculations in a graph counting problem, I have checked that for many values of N (a positive integer), the following is true: $$ \,_2F_1 [\frac{1}{2},-N;-N+ \frac{1}{2} ; z=1] = \frac{4^N (N!)^2}{(2N)!} $$ I would like to prove that this correct for arbitrary N, but I cannot find...

To plot ##u(r)## we need to find the solutions for each region. Which is in the relevant equations part. Now, I have to do this numerically. Using python 3.7 I made an ##u## which is filled with zeros and a for loop with if/elseif statement, basically telling it to plot values for whenever...

Recently I debated with a friend of mine, and the topic was 'Is it reasonable that we call 2(sinx) as trigonometric function?'. My friend said that if we can call y=2(sinx) as trigonometric function, we can call y=x as trigonometric too, because if we call the former 'correct', there is no...

My classmates and I have a big dispute on this question. I think only the first one is right, and they think the third one and the fourth one are right. Please help us to analyze it. Thank you. The original title is as follows ： Which of the following expressions represents y as a function...

Suppose ##\alpha=0##. Then ##\alpha f=0##, the zero map. Hence, the distance between the images of any two ##x_1,x_2 \in D## through ##f##, that is to say, the absolute difference of ##(\alpha f)(x_1)=0## and ##(\alpha f)(x_2)=0##, is less than any ##\epsilon>0## regardless of the choice of...

I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...

Let's say the function is of a form F(x), where F(n) gives us the value of the n-th prime number, for all values of n. And it computes and discovers subsequent prime numbers directly without using any form of brute-force computation.If such a function is discovered, how ground-breaking would it...

Hi, I just need someone to check over my work. I am having trouble with the next part of this question and I just wanted to check that this part was correct first. I have two particles in an infinite square well (walls at x=0 and x=L). I need write an expression for the spatial wave...

Hey everyone . So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me 🤯! It really changed the way I'm thinking ! So i was...

For a lab, I needed to calculate the uncertainty of a refractive index that was found using Snell's law. I found an equation online for propagation of error for any general function, which was I thought that since my equation was I could just get rid of the variable y, and have After...

Question: I have tried this and got work function to e 5.1eV My concern is that for these type of questions, do I need to take into account the signs of some values; such as the negative sign for the charge of an electron? Or could I just take the magnitude for all the values Any help...

I do not know how to proceed.

Hello, I would need some help in calculating the derivative of the function T_el in the attached image. I want to calculate d T_el /d yd, where yd is the variable and it appears in the term I called A_elSide. Its expression is again in the image. Numbers you see are not important.Just to...

So the problem I’m attempting to solve is ##\lim_{x\to a} I_{\alpha}f(x)=\zeta (\alpha )## for f, and a, where ##\zeta (\cdot )## is the Riemann zeta function and ##I_{\alpha}## is the Riemann-Liouville left fractional integral operator, namely the integral equation $$\lim_{x\to...

For 2 bosons each of which can occupy any of the energy levels 0 and E the microstates will be 3 0 E a a aa - - aa the partition function is therefore $$z=1+e^{-\beta E}+e^{-2\beta E}...(1)$$ Another approach to do.. The single particle partition function is $$z=1+e^{-\beta E} $$...

Hey, I am working on a video game in which there will be archers who have the ability to shoot at enemies. My game is two dimensional and I am trying to calculate the angle at which the archer, given an initial velocity, has to shoot in order to hit the target perfectly. I came up with the...

One way to express a function of a matrix A is by a power series (a Taylor expansion). It is not too difficult to show that two functions f(A) and g(A) with such a power series representation must commute, i.e. f(A)g(A) = g(A)f(A). But matrices typically do not commute with their own transpose...

When we calculate the curl of magnetic field, that is the curl of Biot-Savart equation for magnetic field. Please consider these . The working of last equation $$ \nabla \times \mathbf {B} = \frac...

In Wikipedia https://en.m.wikipedia.org/wiki/Associated_Legendre_polynomials, Section Reparameterization in terms of angles, I see this argument: Let ## x = cos\,\theta ## ## \sqrt{1 - x^2} = sin\,\theta ## This is also in Griffiths' Introduction to Quantum Mechanics. Why is this a valid...

I have the function: ##\sqrt{\left(\frac{x}{h}+1\right)^{2}+\left(\frac{y}{h}\right)^{2}}-\sqrt{\left(\frac{x}{h}\right)^{2}+\left(\frac{y}{h}\right)^{2}}## I would like to find an analytical solution, the equivalent function, in the limit of h approaching zero.Additional info which might be...

Φ = hf = 6.626x10^-34x7.5x10^14 = 4.9695x10^-19 J

In a manner analogues to the linearization of functions of a single variable to approximate the value of a function of two variables in the neighbourhood of a given point (x0,y0,z0) where z=f(x,y) using a tangent plane. The tangent plane must pass through the point we wish to approximate z...

Good Morning! If I have function of two variables ##f(x,y)## and if we write it like this $$ z = f(x,y)$$ then it means that for every point in the ##xy## plane there is a point above/below it and is related to it by ##f##. In simple words, every point in ##xy## plane has a point associated...

I've seen position as a function of time in Newtonian physics and potential energy as a function of position, is there an inverse? Any instance where position is a function of energy eg KE, PE. Maybe this is more appropriate for quantum mechanics or modern physics.

I thought I could start somewhere along the lines of ##\psi(x,t)= \psi(x,0)e^{-iE_nt/\hbar}##, but I'm not sure what ##E_n## would be. I also thought about doing the steps listed below in the picture, but I'm not sure how to decompose ##\psi(x,0)## like it says to in the first step. Any help...

Does it make sense to just talk about the average value of a function without specifying the range over which the average is taken? It seems a common occurrence in discussions of waves to just mention that the average value of the complex exponential ##e^{ix}## is zero. But it will be zero only...

Suppose I have an exact microscopic distribution function in phase-space defined as a sum of delta-functions, i.e $$F( \mathbf x, \mathbf v, t) = \sum_{i} \delta( \mathbf x - \mathbf x_i ) \delta (\mathbf v - \mathbf v_i )$$ Can I conclude that, in absence of creation/destruction of particles...