What is Series expansion: Definition and 186 Discussions
In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.
There are several kinds of series expansions, such as:
Taylor series: A power series based on a function’s derivatives at a single point.
Maclaurin series: A special case of a Taylor series, centred at zero.
Laurent series: An extension of the Taylor series, allowing negative exponent values.
Dirichlet series: Used in number theory.
Fourier series: Describes periodical functions as a series of sine and cosine functions. In acoustics, e.g., the fundamental tone and the overtones together form an example of a Fourier series.
Newtonian series
Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.
Zernike polynomials: Used in optics to calculate aberrations of optical systems. Each term in the series describes a particular type of aberration.
Stirling series: Used as an approximation for factorials.
we know that we can expand the following function in Legendre polynomials in the following way
in the script given yo us by my professor, ##\frac 1 {|\vec x -\vec x'|}## is expanded using geometric series in the following way:
However, I don't understand how ##\frac 1 {|\vec x -\vec...
I am trying to understand the series expansion of $$(1-cx)^{1/x}$$ The wolframalpha seems to solve the problem by using taylor series for ## x\rightarrow 0## and Puiseux series for ##x\rightarrow \infty##. Any ideas how can I calculate them ...
Hello,
I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused.
Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
I want to check my calculations via mathematica.
In the book I am reading there's this expansion:
$$\frac{(1+\frac{1}{j})^x}{1+x/j}=1+\frac{x(x-1)}{2j^2}+\mathcal{O}(1/j^3)$$
though I get instead of the term ##\frac{x(x-1)}{2j^2}## in the rhs the term: ##-\frac{x(x+1)}{2j^2}##.
So I want to...
So here's my homework question:
This is the reference formula along with the Rung-Kutta form with the variables mentioned in the question
Here is my attempt so far:
Problem is that i am unsure how to expand this to even get going. I tried referencing my text Math Methods by Boas which has...
Hello Everyone!
I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...
a) I think I got this one (I have to thank samalkhaiat and PeroK for helping me with the training in LTs :) )
$$\eta_{\mu\nu}\Big(\delta^{\mu}_{\rho} + \epsilon^{\mu}_{ \ \ \rho} +\frac{1}{2!} \epsilon^{\mu}_{ \ \ \lambda}\epsilon^{\lambda}_{ \ \ \rho}+ \ ...\Big)\Big(\delta^{\nu}_{\sigma} +...
So I'm on page 67 of Marion/Thornton's "Classical Dynamics of Particles and Systems" and I'm in need of some help. I understand that so far there's is an equation that cannot be solved analytically (regarding motion due to the air resistance and finding the range of the an object shot from a...
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
Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below
I already learn to use Taylor series as:
f(x) = ∑ fn(x0) / n! (x-x0)n
But i don´t see why the serie change when we use differents x0 points.
Por example:
f(x) = x2
to express Taylor series in x0 = 0
f(x) = f(0) + f(0) (x-0) + ... = 0 due to f(0) = (0)2
to x0=1 the series are...
Homework Statement
There is a sawtooth function with u(t)=t-π.
Find the Fourier Series expansion in the form of
a0 + ∑αkcos(kt) + βksin(kt)
Homework Equations
a0 = ...
αk = ...
βk = ...
The Attempt at a Solution
After solving for a0, ak, and bk, I found that a0=0, ak=0, and bk=-2/k...
Homework Statement
Homework EquationsThe Attempt at a Solution
[/B]
Hi,
I am trying to understand the 2nd equality .
I thought perhaps it is an expansion of ##(1-\frac{z}{w})^{-2}## (and then the ##1## cancels with the ##1## in ##( (1-\frac{z}{w})^{-2}) -1 ) ##) in the form ##(1-x)^{-2}##...
Homework Statement
Show that ##\displaystyle \frac{1}{1+x^2} = \frac{1}{x^2} - \frac{1}{x^4} + \frac{1}{x^6} - \frac{1}{x^8} + \cdots##
Homework EquationsThe Attempt at a Solution
I know that the power series expansion of ##\displaystyle \frac{1}{1+x^2}## about ##x=0## is ##1-x^2 + x^4 - x^6 +...
Hi, I stamped at a series expansion. It is probably Taylor. Would you explain it? It's in the vid.
https://confluence.cornell.edu/display/SIMULATION/Big+Ideas%3A+Fluid+Dynamics+-+Differential+Form+of+Mass+Conservation
I understand equation 1 in the picture but I do not understand 2.
I...
<Moderator's note: moved from a technical forum, so homework template missing>
Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
1. ... Expand the Eigenvalue as a power series in epsilon, up to second order:
λ=[3+√(1+4 ε^2)]V0 / 2
Homework Equations
I am familiar with power series, but I don't know how to expand this as one.[/B]The Attempt at a Solution :[/B] I have played around with the idea of using known power...
Homework Statement
Fourier series expansion of a signal f(t) is given as
f(t) = summation (n = -inf to n = +inf) [3/(4+(3n pi)2) ) * e j pi n t
A term in expansion is A0cos(6 pi )
find the value of A0
Repeat above question for A0 sin (6 pi t)
Homework Equations
Fourier expansion is summation...
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...
I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how.
$$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$
I get most of the function, I just can't see where the ##-1## comes from. Could...
Homework Statement
Perform a Taylor Series expansion for γ in powers of β^2, keeping only the third terms (ie. powers up to β^4). We are assuming at β < 1.
Homework Equations
γ = (1-β^2)^(-1/2)
The Attempt at a Solution
I have no background in math so I do not know how to do Taylor expansion...
Homework Statement
I was given a problem with a list of sums of sinusoidal signals, such as
Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion.
Homework Equations
[/B]The Attempt at a Solution
My guess is that it has something to...
Homework Statement
Need to show that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger}
Homework Equations
[a,a^\dagger]=1
The Attempt at a Solution
Need to expand f(a,a^\dagger) in a formal power series. However I don´t know how to do it if the variables don´t commute.
Homework Statement
Find \lim_{x \to 0}\frac{ln(1+x^2)}{1-cos(x)} by using series representations. Check using L'Hospitals rule.
Homework Equations
Taylor polynomial at x=0: \sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}(x)^{k} = f(0) + f'(0)(x) + f''(0)x^{2} +...
The Attempt at a Solution
Using...
Homework Statement
The problem wants me to find the limit below using series expansion.
##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})##
Homework EquationsThe Attempt at a Solution
(1) For startes I'll group the two fractions inside the limit together
##\lim_{x...
I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...
< Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >
hi I've got a problem that I've partially worked but don't understand the next part/have made a mistake?
f(x)=0 for -π<x<0 and f(x)=x for 0≤x≤π
i got a0=π/4 and an=0 and bn=0 if n is even and 2/n if n...
Hi, given the sum of 2 relativistic velocities $ \frac{w}{c} = \frac{\frac{u}{c}+\frac{v}{c}}{1+\frac{uv}{c^2}}$ and setting $ \frac{u}{c}= \frac{v}{c} = (1-\alpha)$, find $ \frac{w}{c} $ in powers of $ \alpha $ through $ {\alpha}^{3} $
I used a binomial expansion to get a tidy intermediate...
Homework Statement
find series expansion of relativistic energy formula, At what speed is there a 10% correlation to the non relativistic KE.
Homework Equations
$$ E=\frac{mc^2}{\sqrt(1- \frac{v^2}{c^2})} = mc^2 + \frac{1}{2}mv^2 + ... $$
The Attempt at a Solution
I found the series above...
Hi, everybody. Mathematic handbooks have given a sum formula for the modified Bessel function of the second kind as follows
I have tried to evaluate this formula. When z is a real number, it gives a result identical to that computed by the 'besselk ' function in MATLAB. However, when z is a...
Homework Statement
"Show that the Hermite polynomials generated in the Taylor series expansion
e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞)
are the same as generated in 7.58*."
2. Homework Equations
*7.58 is an equation in the book "Introductory Quantum Mechanics" by...
Find the Taylor series about a=0 for the function F(x) = \cos(\sqrt{x}).
Taylor series expansion of a function f(x) about a
\sum^{\infty}_0 \frac{f^{(n)}(a)}{n!}(x-a)^n
Taylor series of \cos{x} about a=0 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \ldots
From these...
Homework Statement
"Use power series to evaluate the function at the given point"
## ln (x+ \sqrt{1+{x^2}}) - sin x ##
at ## x = 0.001 ##
Homework Equations
Relevant power series:
A: ## ln (1+x) = \Big( \sum_{n=0}^\infty\frac{({(-1)^{n+1}}{x^n})}{n} \Big) ##
B: ## {(1+x)^p} =...
Homework Statement
Show that the first non-zero coefficient in the expansion of
##e^{-x}-\frac{1-x}{{\left(1-x^2\right)}^{\frac{1}{2}}{\left(1-x^3\right)}^{\frac{1}{3}}}##
in ascending powers of x is that of x^5
Homework Equations
Series expansion, logarithmic series
The Attempt at a Solution...
Homework Statement
Give the first three terms and the general term for ((e^x)-1)/xHomework Equations
None
The Attempt at a Solution[/B]
I wasn't sure if I would be able to take the e^x series and subtract 1 from each term and divide each term by x. I was also thinking of splitting it into...
I'm preparing for my exam and have stumbled across this question. I understand how to execute the l'hopital's rule part of this but I just can't get there. I have no idea how to approach this in order to get a suitable series expansion to the 4th degree of x.
Thank you
This is a very basic question .
Actually in Taylor series expansion of say "sin x" we write the expansion ... (as it is,I am not writing it)
But when we are asked to write the expansion of sin(x^2) we just replace 'x' by "x^2" in the expansion of sin x.
Or if asked some other function such as...
Homework Statement
Let's pretend I am given a potential energy function and nothing else. I need to find the effective spring constant for oscillation about the equilibrium point using a taylor series expansion. I can't find an example or explanation anywhere on how to do this. the potential...
Homework Statement
Find the Laurent series expansion of f(z) = \log\left(1+\frac{1}{z-1}\right) in powers of \left(z-1\right).
Homework Equations
The function has a singularity at z = 1, and the nearest other singularity is at z = 0 (where the Log function diverges). So in theory there should...
find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence.
i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?
Hi all. I'm working on a project that requires me to perform calculations in Fermi normal coordinates to certain orders, mostly 2nd order in the distance along the central worldline orthogonal space-like geodesics. In particular I need a rotating tetrad propagated along the central worldline...
Hi,
I just read a physics paper and there it expands signum of a sine function as below:
sgn(sin(wt))=(4/pi)*{sin(wt)+(1/3)*sin(3wt)+(1/5)*sin(5wt)+...}
How can we expand sgn(sin(wt)) like this?
Thanks.
Hello everyone,
I am currently reading chapter two, section 3 of Griffiths Quantum Mechanics textbook. Here is an excerpt that is giving me some difficulty:
"Formally, if we expand V(x) in a Taylor series about the minimum:
V(x) = V(x_0) + V'(x_0) (x-x_0) + \frac{1}{2} V''(x_0)(x-x_0)^2...
Homework Statement
Problem:
Find the first eight coefficients (i.e. a_0, a_1, a_2, ..., a_7) of the power series expansion
y = ##Σ_{n = 0}^{∞}## [##a_n## ##x^n##]
of the solution to the differential equation
y'' + xy' + y = 0
subject to the initial-value conditions y(0) = 0, y'(0)...
Homework Statement
f = \frac{1}{z(z-1)(z-2)}
Homework Equations
Partial fraction
The Attempt at a Solution
R1 = 0 < z < 1
R2 = 1 < z < 2
R3 = z > 2
f = \frac{1}{z(z-1)(z-2)} = \frac{1}{z} * (\frac{A}{z-1} + \frac{B}{z-2})
Where A = -1 , B = 1.
f = \frac{1}{z} *...
Homework Statement
Find the n+1 and n-1 order expansion of \stackrel{df}{dy}Homework Equations
(n+1)Pn+1 + nPn-1 = (2n+1)xPn
ƒn = \sum CnPn(x)
Cn = \int f(x)*Pn(u)The Attempt at a Solution
I know you can use the recursion relation for Legendre Polynomials once you combine Cn with the...
Show that for small positive $x$, $$\left( \sin x \right)^{\cos x} = x -\left( 3 \log x + 1\right) \frac{x^{3}}{3!} + \Big( 15 \log^{2} x + 15 \log x + 11 \Big) \frac{x^{5}}{5!} + \mathcal{O}(x^{7})$$