Hello guys,
I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work...
3) Taylor expansion question in the context of Lie algebra elements:
Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g(
\alpha)...
I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
Homework Statement
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From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p.
Homework Equations
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I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...
Hi,
Before I post my question, let me admit that my foundation on mathematics is poor. I am trying to work on it, specifically on the application part.
When I came through the following image, I was stuck to understand why I will need one like Taylor's series in a simple case like "F+ΔF = F...
In this derivation:
https://cpb-us-e1.wpmucdn.com/sites.northwestern.edu/dist/8/1599/files/2017/06/taylor_series-14rhgdo.pdf
they assume in equation (8) that x >> a in order to use the Taylor Expansion because a/x has difficult behavior. Why does that assumption work? Meaning, why can we...
Homework Statement
If I have the following expansion
f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2)
This means for other function U(f(r,t))
U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2)
Then up to...
I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = ao + a1 A + a2A2 .... Also, that the energy doesn't depend on phase, so only even terms will...
Hey everyone
1. Homework Statement
I want to compute the Taylor expansion (the first four terms) of $$f(x) =x/sin(ax)$$ around $$x_0 = 0$$. I am working in the space of complex numbers here.
Homework Equations
function: $$f(x) = \frac{x}{\sin (ax)}$$
Taylor expansion: $$ f(x) = \sum...
Homework Statement
Find the Taylor expansion up to four order of x^x around x=1.
Homework Equations
The Attempt at a Solution
I first tried doing this by brute force (evaluating f(1), f'(1), f''(1), etc.), but this become too cumbersome after the first derivative. I then tried writing: $$x^x...
I recently found out the rule regarding the Taylor expansion of a translated function:
##f(x+h)=f(x)+f′(x)⋅h+\frac 1 2 h^ 2 \cdot f′′(x)+⋯+\frac 1 {n!}h^n \cdot f^n(x)+...##
But why exactly is this the case? The normal Taylor expansion tells us that
##f(x)=f(a)+f'(a)(x-a)+\frac 1...
My textbook doesn’t go into it, can someone tell me why Taylor expansion is used to express spring potential energy? A lot of the questions I do I think I can just use F=-Kx and relate it to U(x) being F=-Gradiant U(x) but I see most answers using the Taylor expansion instead to get 1/2 kx^2...
Homework Statement
Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I...
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...
I have a short doubt: Let f(x) be a fuction that can't be integrated in an analytical way . Is anything wrong if I expand it in a Taylor' series around a point and use this expansion to get the value of the definite integral of the function around that point? Suppose that the interval between...
Consider the function:
$$F(s) =\begin{cases} A \exp(-as) &\text{ if }0\le s\le s_c \text{ and}\\
B \exp(-bs) &\text{ if } s>s_c
\end{cases}$$
The parameter s_c is chosen such that the function is continuous on [0,Inf).
I'm trying to come up with a (unique, not piecewise) Maclaurin series...
Hey guys, I need your help regarding the derivation of the fourth runge kutta scheme.
So, I found http://www.ss.ncu.edu.tw/~lyu/lecture_files_en/lyu_NSSP_Notes/Lyu_NSSP_AppendixC.pdf
this derivation. Maybe you have a clue what tehy are doing in C.54.
So before this they are calculating the...
I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field,
L=½(∂φ)^2 - m^2 φ^2
in the equation,
S[φ]=∫ d4x L[φ]
∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2)
Particularly, it is in the Taylor series...
Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is
$$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
Homework Statement
What's the first order term in the expansion ln(x) about x = 1?
Homework Equations
Taylor series formula
The Attempt at a Solution
The question is multiple choice, and the choices are x, 2x, or (1/2)x. However, when I calculate the first order term in the expansion of ln(x)...
[Note from mentor: this thread was originally posted in a non-homework forum, therefore it does not use the homework template.]
I have been given an equation for the relativistic doppler effect but I'm struggling to see this as a function and then give a first order Taylor expansion. Any help...
i watched a lot of videos and read a lot on how to choose it, but i what i can't find anywhere is, what's the physical significance of the a, if we were to draw the series, how will the choice of a affect it?
Hello friends,
I need to compute the taylor expansion of
$$\frac{x^4 e^x}{(e^x-1)^2}, $$
for ##x<<1##, to find
$$ x^2 + \frac{x^4}{12}.$$
Can someone explain this to me?
Thanks!
Homework Statement
If k is a positive integer, then show that
##\lim_{x\to\infty} (1+\frac{k}{x})^x = \lim_{x\to 0} (1+kx)^\frac{1}{x}##
Homework Equations
L'Hopitals rule, Taylor's expansion
The Attempt at a Solution
How should I begin? Should I prove that both has the same limit, or is...
Let \mathbb{S}^n be a simplex in \mathbb{R}^{n+1}, so \mathbb{S}^{n}=\{x\in\mathbb{R}^{n+1}|\sum{}x_{i}=1\}. Let D be a difference measure on \mathbb{S}^{n} with D(x,x)=0 and x=y for D(x,y)=0. D is also smooth, so differentiable as much as we need.
Let (R) be a convexity requirement for D...
Does it make a sense to define the Taylor expansion of the square of the distance function? If so, how can one compute its coefficients? I simply thought that the square of the distance function is a scalar function, so I think that one can write
$$
d^2(x,x_0)=d^2(x'+(x-x'),x_0)=d^2(x',x_0) +...
I thought about the Taylor expansion on a Riemannian manifold and guess the Taylor expansion of ##f## around point ##x=x_0## on the Riemannian manifold ##(M,g)## should be something similar to:
f(x) = f(x_0) +(x^\mu - x_0^\mu) \partial_\mu f(x)|_{x=x_0} + \frac{1}{2} (x^\mu - x_0^\mu) (x^\nu -...
Homework Statement
## L (v^2 + 2 \pmb{v} \cdot \pmb{ \epsilon } ~ + \pmb{ \epsilon} ^2)##, where ## \pmb{\epsilon}## is infinitesimal and ##\pmb{v}## is a constant vector (## v^2 ## here means ## \pmb{v} \cdot \pmb{v} ## ), must be expanded in terms of powers of ## \pmb{\epsilon} ## to give...