Questions on dealing with series data and constructing power series expansions of functions.

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11
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0answers
230 views

Why does $\frac{\partial}{\partial x}O\left(\left(\frac{1}{x}\right)^0\right)$ equal $O\left(\left(\frac{1}{x}\right)^0\right)$ in a series expansion?

When taking the derivative of a series expansion around a finite point, the $O(x^n)$ part is differentiated as expected. $O(x^n)$ becomes $O(x^{n-1})$ except $O(x^0)$ which stays $O(x^0)$. When ...
9
votes
0answers
358 views

How does Mathematica find a series expansion of expressions containing logarithms when there is a singularity at the expansion point?

I am looking for a good approximation to a function containing logarithms, especially at values close to zero. When I used Mathematica's Series command I found ...
6
votes
0answers
111 views

Design considerations behind `O` (a.k.a. BigOh, a.k.a. Landau Order)

This works without any warnings: O[Log[x]]. This raises a warning: O[x^2]. I have a few questions around this: Why is it a ...
5
votes
0answers
113 views

Expansion of $E(i c \mid m)$ at $c\to\infty$?

Currently, I am using a Windows machine with Mathematica 8. I noticed a difference in a series expansion of the function EllipticE[] in comparison with a result ...
4
votes
0answers
53 views

Quirky behavior of Series[]

This is a "what's going on?" question about MMa behavior, not so much a "how to fix?" This code calculates a Taylor series for a two-term Gaussian Mixture: ...
4
votes
0answers
119 views

Proper treatment of roots and powers in Series?

I have the following problem in Mathematica 9 on Linux. I let Mathematica compute the Series expansion: ...
3
votes
0answers
93 views

Discrepancy in the series expansion of $\log(z/(z-1))$ in Mathematica 5

When I calculate the series expansion of $\log\frac{z}{z-1}$ around $0$ in Mathematica 5.2, I obtain: ...
3
votes
0answers
142 views

Generating function for Newton series?

The function GeneratingFunction gives generating function for Taylor series. Is there a similar function for Newton's series? $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)$$
2
votes
0answers
43 views

Peculiarities with Series and fractional exponents or bug?

The documentation of the Series[] function states that it can handle "certain expansions involving negative powers, fractional powers, and logarithms." What are the ...
2
votes
0answers
55 views

Puiseux series for algebraic curves

Has anyone implemented a function in Mathematica that computes Puiseux expansions of algebraic curves? Using something like ...
2
votes
0answers
54 views

Implementation of Series function

I'm trying to evaluate the partial derivatives of some function F[x,y] at some point, i.e. ...
2
votes
0answers
79 views

Why does Mathematica not go to specified order in series?

For some reason Mathematica will not evaluate this asymptotic series to the requested order. Inputting: ...
2
votes
0answers
136 views

Series expansion: Taylor series takes huge amount of time

I'm working on a notebook, trying to expand the root of a cubic polynomial in Taylor series. When I type: ...
2
votes
0answers
138 views
2
votes
0answers
142 views

Changing bounds of summation after differentiating symbolic sums

Suppose, I have a function written as Taylor-Maclaurin series f = Sum[c[n]*x^n, {n, 0, Infinity}] Now, I wish to differentiate this expression with respect to ...
1
vote
0answers
102 views

Asymptotics: replacing all arguments in a trig function with dummy variable after manipulation

I'm doing manipulations on functions of the form $$\phi(x,z,t) = \sum_n a_n(\epsilon x,\epsilon t,z)e^{in( k_ox -\omega_o t + \epsilon \theta(\epsilon x,\epsilon t)))} e^{(k_o+\epsilon k(\epsilon ...
1
vote
0answers
133 views

Best way to determine polynomial coefficients in series expansion

I would like to solve the equation $$h'(\boldsymbol{x}_1)\left[B_1\boldsymbol{x}_1+g_1(\boldsymbol{x_1},h(\boldsymbol{x}_1))\right]=B_2h(\boldsymbol{x}_1)+g_2(\boldsymbol{x}_1,h(\boldsymbol{x}_1))$$ ...
1
vote
0answers
66 views

Decimal representations of analytic values in an expansion

I have a solution to an ODE (which I call sol), which I would like to expand in terms of Log[1+x] around x=-1. I thought that: ...
0
votes
0answers
10 views

Using InverseSeries on series with non symbol series parameter

I am attempting to use InverseSeries in the following way: InverseSeries[A[y]+A[y]^2+O[A[y]]^3, A[x]] However, upon execution it does not perform the operation, ...
0
votes
0answers
43 views

Dirichlet series expansion

Is there a way to get Dirichlet series expansions in Mathematica? For example, I would want DirichletSeries[Zeta[s], {s, 4}] to return ...
0
votes
0answers
42 views

Constructing a function for expanding general $n$ products

I have the following quantum mechanically motivated product: $\langle0\vert(A_1b_1 + A_2b_2 + A_3b_3)(B_1b_1 + B_2b_2 + B_3b_3)(C_1b_1 + C_2b_2 + C_3b_3)$, where $b_i$ is an annihilation operator ...
0
votes
0answers
56 views

Series of square root of exponential

my problem concerns the Series command applied to Sqrt of an exponential, and it can be presented in a simplified version as ...
0
votes
0answers
32 views

Subtracting Series

When I input the following $\sum_{n=0}^{m+1}x[n]-\sum_{n=0}^{m}x[n]$ which in InputForm is: Sum[x[n], {n, 1, 1 + m}] - Sum[x[n], {n, 1, m}] it returns ...
0
votes
0answers
31 views

Don't understand Series expansion in my scope

I have a quite complex, implicit problem where I have the following (short version): ...
0
votes
0answers
32 views

FourierSinCoefficients extremely slow

I'm just playing with Fourier decomposition of periodic functions. We had a similar example with standing waves and cosines in class, so I tried to expand a triangle wave in sines. This is what I came ...
0
votes
0answers
64 views

Linearization of differential equation - need tips for the use of Series function

I would like to linearize a differential equation around a equilibrium position. The description of the steps that I have carried out are : Equation ...
0
votes
0answers
109 views

Alternative to Series

Here is a sample of my code: ...
0
votes
0answers
32 views

SumConvergence with product $\sum_1^\infty{\frac{1\times3\times5\times…\times(2n-1)}{n!}}$

SumConvergence[( Product[(2 n - 1), {n, 1, infinity}])/n!, n] $$\sum_1^\infty{\frac{1\times3\times5\times...\times(2n-1)}{n!}}$$ However this returns true but it ...
0
votes
0answers
50 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
0
votes
0answers
191 views

A power series expansion

Consider the function, $f(z) = z tanh(\pi z) log (z^2 + a^2)$ for some $a>0$. Now I am considering 3 different situations, $z = i(n+0.5) - i\epsilon + \delta - it$ for $n \in \mathbb{Z}$ and ...
0
votes
0answers
462 views

Solving differential equations with sums (power series)

I have sets of 10 differential equations, but for this purpose I'll demonstrate what I need on one example that can be solved by hand. My equation is this: ...
0
votes
0answers
55 views

Degeneracy problem at the start of a series expansion

I am facing the following problem : I have an equation which could write $$ F(x_1(Y))-F(x_2(Y)) = 0 $$ $x_1$ and $x_2$ are the largest and smallest roots of a cubic equation; their definition ...