Approximation Definition and 705 Threads

Hello! (Wave) I want to prove that each continuous function $f$ in a closed and bounded interval $[a,b]$ can be approximated uniformly with polynomials, as good as we want, i.e. for a given positive $\epsilon$, there is a polynomial $p$ such that $$\max_{a \leq x \leq b} |f(x)-p(x)|<...

Homework Statement A plane z=0 is charged with density, changing periodically according to the law: σ = σ° sin(αx) sin (βy) where, σ°, α and β are constants. We have to find the potential of this system of charges. Homework EquationsThe Attempt at a Solution [/B] I...

Hi there. I have a question about the damped pendulum. I am working on an exercise where I have already numerically approximated the solution for a simple pendulum without dampening. Now, the excercise says that I can simply change the code of this simple situation to describe a pendulum with...

I'm preparing for an exam and I expect this or a similar question to be on it, but I'm running into problems with using the Born approximation and optical theorem for scattering off of a finite well. 1. Homework Statement Calculate the cross sectional area σ for low energy scattering off of a...

Homework Statement Assume that Planck's constant is not actually constant, but is a slowly varying function of time, $$\hbar \rightarrow \hbar (t)$$ with $$\hbar (t) = \hbar_0 e^{- \lambda t}$$ Where ##\hbar_0## is the value of ##\hbar## at ##t = 0##. Consider the Hydrogen atom in this case...

Homework Statement For a single mechanical unit lung, assume that the relationship among pressure, volume, and number of moles of ideal gas in the ling is given by PA((VL)/(NL)a = K, where a = 1 and K is a constant. Derive the lowest-order (linear approximation to the relationship among changes...

Homework Statement Equation (10.30) in Jackson is the first-order Born approximation. What is the second-order Born approximation? Homework EquationsThe Attempt at a Solution I can get the first-order Born approximation in Jackson's textbook. If I want to obtain the second-order (or n-th...

Homework Statement Suppose we have: ## f(x) = x^2 - b ## ## b > 0 ## ## x_0 = b ## And a sequence is defined by: ## x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i) } ## prove ## \forall i \in N ( x_i > 0 ) ## Homework Equations The Attempt at a Solution a)Here I tried solving for ## x_1 ## as...

Homework Statement An oscillator when undamped has a time period T0, while its time period when damped. Suppose after n oscillations the amplitude of the damped oscillator drops to 1/e of its original value (value at t = 0). (a) Assuming that n is a large number, show that...

I'm unsure on how to start this problem. I tried to make a tree diagram but to no avail did it help out. Question: On average, Mike Weir scores a birdie on about 20.9% of all the holes he plays. Mike is in contention to win a PGA golf tournament but he must birdie at least 4 holes of the last 6...

Hey! :o We have \begin{equation*}A:=\begin{pmatrix}-5.7 & -61.1 & -32.9 \\ 0.8 & 11.9 & 7.1 \\ -1.1 & -11.8 & -7.2\end{pmatrix} \ \text{ and } \ z^{(0)}:=\begin{pmatrix}1\\ 1 \\ 1\end{pmatrix}\end{equation*} I want to approximate the biggest (in absolute value) eigenvalue of $A$ with the...

Homework Statement Homework EquationsThe Attempt at a Solution Applying conservation of potential energy, ## mgL (1 - \cos{ \theta_0}) = mg(L + \delta ) (1 - \cos{ \theta_1}) ## ## \cos{ \theta_1} - \cos{ \theta_0} = \frac { \delta - \delta \cos{ \theta_1}} L ##Taking the...

Homework Statement The number of flaws in a plastic panel used in the interior of cars has a mean of 2.2 flaws per square meter of panel . What's the probability that there are less than 20 surface flaws in 10 square meter of panel ? Homework EquationsThe Attempt at a Solution This is a...

Hi, as you can see at the end of the picture/attached file collision integral is approximated to a discrete sum. Could you express how this approximation is derived?

Equation \varphi(x)=x+1-\int^{x}_0 \varphi(y)dy If I start from ##\varphi_0(x)=1## or ##\varphi_0(x)=x+1## I will get solution of this equation using Picard method in following way \varphi_1(x)=x+1-\int^{x}_0 \varphi_0(y)dy \varphi_2(x)=x+1-\int^{x}_0 \varphi_1(y)dy \varphi_3(x)=x+1-\int^{x}_0...

Homework Statement In the attachments there is the question and its solution, it's problem 3.5. Homework EquationsThe Attempt at a Solution My question is how did they get the dimensionless Hamiltonian in both cases, and how did they explicitly calculated ##m## in both cases? I assume it's...

Homework Statement The following question and its solution is from Bergersen's and Plischke's: Equation (3.38) is: $$m = \frac{\sinh (\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}$$ Homework EquationsThe Attempt at a Solution They provide the solution in their solution manual which I...

Homework Statement Homework EquationsThe Attempt at a Solution I don't see how to do this calculation of ##Z_c##, I need somehow to separate between ##\sigma_j=1## and ##\sigma_j=-1##, and what with ##\sigma_0##?

I used Newtons method and taylor approximations to solve this equation $$f'''+\frac{m+1}{2}ff''+m(1-f^{'2})=0$$ It solves for velocity of air over a flat plate. The velocity is a constant ##u_e## everywhere except in a boundary layer over the plate, where the velocity is a function of distance...

Hello everyone, In Born-Oppenheimer approximation there is one step, when you divide your wavefunction into two pieces - first dependent on nuclei coordinates only and second dependent on electron coordinates only (the nuclei coordinates are treated as parameter here). The "global"...

Homework Statement A ball is dropped from rest at height ##h##. We can assume that the drag force from the air is in the form ##F_d=-m \alpha v##. I know then the position in function of the height $$y(t)=h-\frac{g}{\alpha} (t-\frac{1}{\alpha} (1 - e^{-\alpha t}))$$ If I take ##\alpha t<<1##...

Homework Statement [/B] It's been a couple of years since differential equations so I am hoping to find some guidance here. This is for numerical analysis. Any help would be much appreciated. Homework EquationsThe Attempt at a Solution

Homework Statement Find the quadratic least squares Chebyshev polynomial approximation of: g(z) = 15π/8 (3-z^2)√(4-z^2) on z ∈ [-2,2] Homework Equations ϕ2(t) = c0/2 T0(t) +c1T1(t)+c2T2(t) T0(t)=1 T1(t)=t T2(t)=2t2-1 Cj = 2/π ∫ f(t) Tj(t) / (√(1-t2) dt where the bounds for the integration...

Hello, My name is Hugh Carstensen. I am a CSE undergrad at the Ohio State University. I recently secured a position designing and assembling an automated camera-rig for digitization of archival works in the Knowlton School of Architecture. The rig will be powered by a number of small stepper...

Homework Statement I want to solve the motion equation ## m \frac {dv_z} {dt} = - μ \frac {∂B_z} {∂z} ## with small angle approximation Homework Equations ## B_z(z) = B_0 -bCos(\frac {zπ} {2L}) ## is the magnetic field in the z-direction The Attempt at a Solution Started by derive the...

In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls...

In the following equation, $$P(x; a)= \frac{\gamma}{2\lambda L \eta} [\frac{1}{π^2N_F(a)\eta(1 - \frac{x}{a\eta})^2} + \frac{1}{π^2N_F(a)\eta(1 + \frac{x}{a\eta})^2} +\frac{2}{π^2N_F(a)\eta(1 - \frac{x^2}{a^2\eta^2})} [sin (\frac{π N_F(a)\eta(1 - \frac{x}{a\eta})^2}{2})sin (\frac{πN_F(a)\eta(1...

Homework Statement Approximate ##~\sqrt[4]{17}~## by use of differential Homework Equations Differential: ##~dy=f(x)~dx## The Attempt at a Solution $$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$ $$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$...

Homework Statement In the Griffiths book <Introduction to QM>, Section 2.3.2: Analytic method (for The harmonic oscillator), there is an equation (##\xi## is very large) $$h(\xi)\approx C\sum\frac{1}{(j/2)!}\xi^{j}\approx C\sum\frac{1}{(j)!}\xi^{2j}\approx Ce^{\xi^{2}}.$$ How to understand the...

Say whether each statement is TRUE OR FALSE. Do not use a calculator or tables; use instead the approximations sqrt{2} is about 1.4 and π is about 3.1. 1. 2 < or = (π + 1)/2 2. sqrt{7} - 2 > or = 0 For question 1, I replace π with 3.1, and then simplify, right? How do I apply the...

Homework Statement Hy guys I am having an issue with approximating this first question, which I have shown below. Now my problem is not so much solving it but I have been thinking that if given the same question without knowing that it approximates to so for example the question I am...

I'm following this video: The professor says that for small angles, tan(Θ) = dy/dx. I don't understand why this is so. Tan(Θ) is equal to sin(Θ) / cos(Θ), and if Θ is small, then cos(Θ) is about 1, which means dx = 1, not a infinitesimally small number.

Homework Statement Homework EquationsThe Attempt at a Solution So cts approx holds because ##\frac{E}{\bar{h}\omega}>>1## So ##\sum\limits^{\infty}_{n=0}\delta(E-(n+1/2)\bar{h} \omega) \approx \int\limits^{\infty}_{0} dx \delta(E-(x+1/2)\bar{h}\omega) ## Now if I do a substitution...

According to WKB approximation, the wave function \psi (x) \propto \frac{1}{\sqrt{p(x)}} This implies that the probability of finding a particle in between x and x+dx is inversely proportional to the momentum of the particle in the given potential. According to the book, R. Shankar, this is...

<Moved from a technical section and thus a template variation> 1-) Question: Let f, g and h be differentiable everywhere functions with h(1) = 2 , h'(1) = - 3 , g(2) = -1 , g'(2) = 5 , f(-1) = 4 , f'(-1) = 7. Approximate the value of function F(x) = f(g(h(x))) at point x= 1.001 2-) My...

So we are studying optics in school this semster, Very interseting topic I say but I just have a couple of question I want to ask. In concave and convex mirror, we study spherical ones where F = R/2. I was able to prove this and that it is only an approximation when ## R >> h_o ## or ## h_0##...

Given a unit-hypotenuse triangle, how do we get the inverse sin/cos/tan equations? I'm trying to program a high-precision fixed-fraction model of the sun and Earth and I've forgotten how the equations are derived. I know there's differentiation and integration. And I'm stuck on how to express...

I've arrived at it not by using some mainstream mathematics. I'm looking for a proof which involves some widely-known mathematics. I'm sorry if I'm using my own notation, but it's the only way to make the expression compact. The notation is: $$log^n_xy$$: For log with the base x applied n times...

Homework Statement Show that, for an extremely relativistic particle, the particle speed u differs from the speed of light c by $$ c - u = (\frac {c} {2}) (\frac {m_0 c^2} {E} )^2, $$ in which ##E## is the total energy.Homework Equations I'm not sure what equations are relevant. This...

Homework Statement Spherical,homogeneous star with radius R orbiting black hole at distance ## r_p >>R ## .Derive the tidal force acting upon the star by dividing the star into two equal parts and making the necessary approximations. Homework Equations The tidal force equation of ## a \propto...

Hello. A whole decade passed since I graduated mathematics and shifted to other profession, so my knowledge is very rusty. There is an important problem for a scientific work that I need help for. Let's say factor t is being calculated from factors x, y and z, all some parameters from living...

Text books ordinarily give the escape velocity of a mass-M body (in the center of mass frame of the system of the body and the escaping projectile, whose mass I'll label m) as (*) v2 = 2GM/r where r is the distance between the body and the escaping projectile. it doesn’t seem to me that (*)...

Could you tell me the reason that if pole is close to the imaginary axis, (1) can be same as (2).

I came across a guy claiming that the "best approximation" for the natural logarithm of a number is this: ln x=2^n*(x^(2^-n)-1) Oddly enough, it seems to work rather well! I don't really get why it does... I also don't know if it has a limit, I couldn't test it as I don't have access to my...

Homework Statement The Schrodinger equation is given by $$i\hbar\ \frac{\partial}{\partial t}\ \mathcal{U}(t,t_{0})=H\ \mathcal{U}(t,t_{0}),$$ where ##\mathcal{U}(t,t_{0})## is the time evolution operator for evolution of some physical state ##|\psi\rangle## from ##t_0## to ##t##.Rewriting...

Homework Statement Homework Equations The Attempt at a Solution I think this problem is probably a lot simpler than I am making it out to be. However, when I apply sterling's approx., I get a very nasty equation that does not simplify easily. One of the biggest problems I have though is...

Homework Statement A point particle of mass m slides without friction within a hoop of radius R and mass M. The hoop is free to roll without slipping along a horizontal surface. What is the frequency of small oscillations of the point mass, when it is close to the bottom of the hoop...

Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...

I am solving the simple 2nd-order wave equation: $$ \frac {\partial ^2 E}{\partial t^2} = c^2 \frac {\partial ^2 E}{\partial z^2} $$ Over a domain of (in SI units): ## z = [0,L=10]##m, ##t = [0,t_{max} = 10]##s and boundary/initial conditions: $$ E(z=0) = E(z=L) = 0 $$ $$ E(t=0) =...

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...