Function Definition and 1000 Threads

I want to find the analytical solution to the integral given below. \int_{-\infty}^{\infty} \frac{ sinc^2(\frac{k_yb}{2})}{\sqrt{k^2 - k_x^2 - k_y^2}}dk_y In other words, \int_{-\infty}^{\infty} \frac{ \sin^2(\frac{k_yb}{2})}{(\frac{k_yb}{2})^2\sqrt{k^2 - k_x^2 - k_y^2}}dk_y Can this be...

Hello, I want to know what is the incresing and decreasing interval of this even function $|e^x+e^{-x}|?$ If any member knows the correct answer, may reply to this question.

I am not sure about finding the limit of the integral when it comes to finding the CDF using the distribution function technique. I know that support of y is 0 ≤y<4, and it is not a one-to-one transformation. Now, I am confused with part b), finding the limits when calculating the cdf of Y...

Let $f$ be differentiable from $(-\inf,0)$ to $(0,\inf)$ and let $f'(x)<0$ for all real numbers except 0 and $f'(0)=0$. Prove that f is strictly decreasing.

I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing...

Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...

Weierstrass function is the classic example of a continuous function which is nowhere differentiable. What happens when a function is monotone? My guess that it cannot be nowhere differentiable. It seems to me the reverse is true - it is differentiable almost everywhere. Any light on the...

I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...

Hi, I have this formula, What I want is to find the value of "x" (without trying all possibilities) so that the result of the formula will be the lowest possible value under the constraint when x !=0, and x<n. Here, values of A,B,C, Q, R,n are already known and fixed...

I'm trying to simulate a simple series reaction stochastically using Gillespie's algorithm. I found this file: What is this 'propensity function'? Say for example I have the simple reactions: A --(k1)--> R R--(k2)--> S are these 'propensity functions' the rates (a wild guess)? I mean; α1 =...

The movement in the z-direction is easy to solve for, as it's only affected by the gravitational force. However, if there's a magnetic field pointing down along the z-axis, the particle is going to be accelerated along the y-axis (F=q*v *B). The force is always going to be perpendicular to the...

I struggle to find an appropriate inverse Laplace transform of the following $$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$ WolframAlpha gives as an answer $$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$ which...

double foo(int arr[], double *ave, int index){ double *s; *s=*ave; // calculation// return(foo (arr,ave,index)); // other calculation// } I want to keep the ave value during the recursion, because after ave is calculated, I will do another calculation is recursively in this...

Hello everyone. I try to plot a figure from a journal article. I gave the equations in the inserted image. I wrote the script given below for that. I expect to obtain a plot like the one given on the left but I end up with something totally different. So, the values of ##I_{0}## and ##I_{1}##...

Assumptions: 1) The minimum stellar mass in this cluster is 0.1M⊙ 2) The maximum stellar mass in this cluster is 150⊙ First calculate the local stellar density constant (ξ0) for this cluster using eq 1: Having rearranged this equation and using the limits of the minimum and maximum stellar...

I am looking at Srednicki ch 64 , how does equation 64.1 follow from 64.3 as stated. Explicitly in QED how does ## u_{s'}(p')V^{u}(p',p)u_{s}(p)=e\bar{u'}(F_{1}(q^{2})\gamma ^{u}-\frac{i}{m}F_{2}(q^{2})S^{uv}q_{v})u ## follow from the quantum action ## \Gamma =\int d^{4}x(eF_{1}\bar{\varphi...

I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...

Hey! :o I am looking at the following example of a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ that is not differentiable at any $x\in \mathbb{R}$. For $x\in [-1,1]$ we define $\phi (x)=|x|$ and then we extend $\phi$ to the whole $\mathbb{R}$ such that $\phi (x+2)=\phi (x)$...

I have a problem asking to show that a certain function approaches a quadratic for large values of the variable. And I realize now that this is a skill with which I am totally unfamiliar. Can't use a Taylor series in y= 1/x because the value at y=0 is infinite. Would appreciate a recommended...

I’m using discriminant function analysis to determine the potential accuracy of several biometric measurements being used in conjunction for binary classification purposes for my BSc Biomed research project. Overall I've only got 110 data points so it's a stretch but hey, that's anatomy! What...

If there is no ##(-1)^2## factor, I can find the limit. But, now I have no idea how to find limit for the ##(-1)^\infty##. I thought ##(-1)^\infty## is an indeterminate form. So, how to modify this? Thanks!

Here is a pic of question My attempt-: I defined functions f(x-1) and f(x+1) using f(x).After defining them,I substituted their values in the equation f(x-1)+f(x+1)=sinA. For different ranges of x,I got different equations. For 1<x<2,I got 1-x=sinA. But now I am confused.For each different...

A very simple question but I can't find an answer. I have an expression which depends on two integers, n,d. Now, I want this expression to be a) 1 when d=n=0, b) some expression (that I won't write here) when both d and n are >0 c) zero when wither d or n negative. At first I defined the...

the integral is: and according to mathematica, it should evaluate to be: . So it looks like some sort of Gaussian integral, but I'm not sure how to get there. I tried turning the cos function into an exponential as well: however, I don't think this helps the issue much.

If a polynomial $P(x)=x^3+Ax^2+Bx+C$ has three real roots at least two of which are distinct, prove that $A^2+B^2+18C>0$.

hello , hope all of you are doing well , i have question about the unit of the function of waves of string fixed in both boundary , the function of waves is function of two variables x and t , so it's function describe the displacement in function of place and time , Ψ(x,t)=φ(x)*sin(ωt+α)...

Hello, Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1. I am trying to prove f is injective. Let a,b be in ]1, +inf[ and suppose f(a) = f(b). Then, a^2 + 2a + 1 = b^2 + 2b + 1. How do I solve this equation such that I end up with a = b? Solution: (a + 1) ^2 = (b + 1)^2...

Hello! (Wave) I want to calculate the integral $\int_{-1}^2\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt$. I have done the following so far: $$\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt=\int_{-\infty}^1\sin \left (\pi (t-1)\right )\delta (-t+1)\...

In Sommerfeld’s Lectures on Theoretical Physics, Vol II, Chapter 2, Section 6, Page 43 we derive an expression for the equilibrium of liquids as $$ grad ~p = \mathbf F$$ Where ##p## is the pressure and ##F## is the exertnal force. Then he writes, [ The equation above ]includes a very remarkable...

Given initial displacement ##X_0## and displacement at any time ##t## as ##x##. Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time. For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11...##and so on. From this, is there any method...

Hello all, I am a newcomer here. Not a physicist, just an enthusiast. ;) I was thinking whether it is possible to separate a one-particle wave function into two, "completely disjoint" parts. The following thought experiment explains better what I am thinking about. Let us suppose, that there...

Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist...

Ok all I did was DesmosNot real sure how to take limit

The function is f(x)=x5-12x3+7x2+2x+7. I found the minimum of the function and compared the value to a calculator and it seemed okay. But I am confused as to how to incorporate the interval into my code. Has my code already sufficiently answered the question? from scipy import optimize...

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...