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

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3
votes
1answer
202 views

Is there a function that, given a rational function, will return the general term of its infinite series expansion?

Is here some way to expand a rational function to an infinite sum in Mathematica, i.e., a series? I want the general term of the series. For example, $\dfrac{2}{3(x-1)^3}$
5
votes
2answers
228 views

Symbolically Expanding One-Variable $q$-hypergeometric Series

In general, how do I get Mathematica to symbolically expand various infinite series to an arbitrary degree of accuracy? I deal with $q$-hypergeometric series quite frequently, and specifically want ...
0
votes
1answer
227 views

Asymptotic forms of Bessel function

I want to replace Bessel functions by asymptotic forms, so the question is: can I find the best ones with help of Mathematica? And if it's possible, how can I do it? Update How can I get with ...
2
votes
0answers
35 views

Series expansion in Infinity issue with Zeta(s) function

With this code: Series[Zeta[s], {s, Infinity, #}] & /@ Range[10] // MatrixForm Series expansion for the Zeta(s) ...
3
votes
3answers
153 views

OEIS A144311 Generating function

I'm looking for a way to use calculate OEIS A144311 efficiently in Mathematica. First, let's define the series. In one sense or another, this series considers the number between "relative" twin ...
1
vote
2answers
124 views

How do I get Series[] as a functional, rather than as an expression (i.e. to avoid the dummy variable)?

How can I write an equivalent to Series that doesn't require a dummy variable? Note that the series should be constructed before the evaluation point is supplied, ...
10
votes
3answers
367 views

Chebyshev Approximation

Is there functionality in Mathematica to expand a function into a series with Chebyshev polynomials? The Series function only approximates with Taylor series.
-1
votes
1answer
38 views

Maclaurin Series Help [closed]

My problem is to numerically approximate the series (1 - Cos[x])/x over the interval [0,1]. I typed it into Mathematica as so: ...
1
vote
3answers
53 views

Can highlight an expression in mathematica in an expansion?

I ask mathematica to make an expansion of some expression. Is there a way to ask mathematica to highlight all terms that, say, goes by x^3 since it is really hard to find them? E.g. ...
1
vote
0answers
236 views

A power series expansion [closed]

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 $n \in \mathbb{Z}$ ...
2
votes
0answers
69 views

Ramanujan's asymptotic formula for $p(n)$ [closed]

The following expression is the exact formula of partition function. Ramanujan's asymptotic formula for $p(n)$ is following $$p(n)\sim\frac{1}{4n \sqrt{3}}e^{\pi \sqrt{2n/3}}$$ Can I use ...
4
votes
0answers
80 views

Bug in Series[Pochhammer[1+n, n], {n,Infinity,1}]?

The function Pochhammer[1 + n, n] tends to infinity. We have ...
3
votes
1answer
74 views

Find the function $f(x)$ by using its fourier expansion

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem by using Mathematica How to find the function $f(x)$, if I know its ...
2
votes
0answers
44 views

Unexpected imaginary term in the asymptotic expansion of DawsonF

FullSimplify[Series[2 DawsonF[x], {x, ∞, 8}]] (* -I Exp[-x^2] √π + (1/x + 1/(2 x^3) + 3/(4 x^5) + 15/(8 x^7) + O[1/x]^9) *) What is the reason the term ...
1
vote
1answer
28 views

`Series` acts on variables outside its argument?

I am trying to work with Taylor series. What I am trying to do is to express $$ f(h) - f'(0) = f(h) - f'(h - h) $$ such that only derivatives of $f$ with the argument $h$ appear, for example $$ f(h) - ...
4
votes
1answer
151 views

Why does merely writing SeriesData[] with zeros give an error, even if it is never called?

Bug introduced in 8.0 or earlier and persisting through 10.4.1 Why does the code Function[t, SeriesData[t, 0, {0}, 0, 1, 1]] give me the following error? ...
3
votes
1answer
25 views

Asymptotics of Bessel function for real arguments

I am trying to calculate the following asymptotic behaviour: Normal@Series[BesselK[1, r \[CapitalLambda]] / BesselK[1, \[CapitalLambda]], {r, 0, 1}] but for ...
2
votes
1answer
62 views

Series expansions and algebraic branch points

I have the expression $$(1-x^4)^{-1/4}$$ where $0<x<1$, which gives me a real number. When I try to expand it around $x=1$, Mathematica gives me complex coefficients, which makes me think that ...
4
votes
1answer
116 views

Successive Series Expansion Bug (?)

I've found what appears to be a bug in MMA related to taking successive series expansions. I'm providing this minimal example and post as other posts didn't appear to address the issue I found. In ...
0
votes
2answers
96 views

Solving recursion relation from power series

I am interested in solving differential equations in the form of power series. Let's say we have following equation: $$f^{\prime \prime} (\rho) + \left( \frac{2 e^{-k \rho}}{\rho} - \varepsilon ...
3
votes
0answers
69 views

A series in powers of $(a-z)$ instead of $(z-a)$

Sometimes it is more convenient to find a series expansion (e.g., Taylor, Laurent, Puiseux, ...) in powers of $(a-z)$ than in powers of $(z-a)$. For instance, the command ...
0
votes
0answers
56 views

break down the function

I am using Series function, but I found that it cost too much time. Since the Series function is Taylor expansion and I only need first order, I want to break down the code, so I can make it faster ...
0
votes
0answers
53 views

Sum of Infinite convergent series

I have a series as Sum[(x^1 - (x - 1)^1) (b^Log[(1 + (x))]), {x, 1, Infinity}] which is convergent when b=0.3 and ...
1
vote
2answers
62 views

How to expand an infinite product?

How can I use Mathematica to expand such a product (only need a finite number of terms): $$\prod^{\infty}_{n=1}\frac{({1-yq^{n+1}})({1-y^{-1}q^n})}{(1-q^n)^2}$$
1
vote
1answer
114 views

Problem with solving a system of differential equations [closed]

I am trying to reproduce a result that is part of a derivation of the flow due to a rotating disk. I have as given this system of differential equations. (In these equations primes indicate ...
2
votes
3answers
137 views

How to write the Taylor expansion of the sum of several matrices?

When $A$ and $B$ are matrices, we have the following Taylor expansion for the inversion function together with basic information on convergence: $$ ...
8
votes
3answers
179 views

Expand series unevaluated

I'm newbie in Mathematica. I'd like to obtain nice and verbose output for any series calculation. For example, given a simple sequence n*(-1)^(n-1) and ...
2
votes
1answer
49 views

Getting rid of sub-exponential terms in an asymptotic expansion for a modified Bessel function

I am trying to get Mathematica to produce suitable asymptotic expansions for some modified Bessel functions at large argument (more specifically, the expansion in the DLMF's eq. (10.40.1)), and I'm ...
3
votes
0answers
84 views

How to force Series[] to compute expansions by considering non commutative multiplication?

I wish to compute the Taylor series expansion of the following iteration method $x_{k+1}=x_k-f'(x_k)^{-1}f(x_k)$ up to four terms of error ($e_k=x_k-\alpha$). When this is a scalar iteration, I very ...
0
votes
1answer
52 views

How to evaluate and simplify series expansions of long partial derivatives involving trigonometry?

I have 4 maps defined on a plane, i.e. on $\mathbb{R}^{2}$. Let's call them $F,G,H,J$. Let $(s,t) \in \mathbb{R}^{2}$. Define $F(s,t) = (f_{1}(s,t), f_{2}(s,t)).$ Likewise, in components, write $$ ...
0
votes
0answers
73 views

Using Mathematica to derive an equation

My goal is to obtain Eq. (5) given in this Paper using Mathematica. First we need to define Eqs. (2) and (3) ...
0
votes
2answers
465 views

Expansion in Basis Functions

I am trying to create an expansion of the form f[x,y]->Sum[Cn[x] y^n,{n,1,order}] To replace the function f by the ...
2
votes
0answers
32 views

Polynomial kernel expansion

I am trying to calculate the polynomial kernel expansion using Mathematica. I have tried the Expand and Simplify functions with ...
1
vote
2answers
67 views

Engel expansion

How do I do an Engel expansion in Mathematica? And find the unique non-decreasing sequence of positive integers $ \{ a_1 ,a_2,a_3,\dots \}$ , such $$\frac{e}{\pi}=\frac{1}{a_1}+\frac{1}{a_1 ...
5
votes
5answers
161 views

How to find the derivative value at $(\pi,0)$ for this implicit function $n$ times?

I am trying to take the implicit derivative at $\sin(x+y)+\sin(x)=y$ and substitute $x=\pi$ and $y=0$ at least 6-7 times since I need to find the Taylor series for this function. Since I barely ...
26
votes
5answers
7k views

Multivariable Taylor expansion does not work as expected

The basic multivariable Taylor expansion formula around a point is as follows: $$ f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
20
votes
4answers
600 views

Expansion of a hypergeometric function takes ages with Mathematica 9 and 10 (regression?)

Mathematica 8 (Linux version) can evaluate AbsoluteTiming[Series[Hypergeometric2F1[1, 1 - eps/2, 3 - eps, 1/2], {eps, 0, 0}]] in no time. On one of the ...
1
vote
1answer
32 views

Multivariables series expansion up to some power of all the variables [duplicate]

I have a function f[x, y, z] that I would like to expand up to a given power of xyz. For now, I am using ...
0
votes
1answer
80 views

How can I calculate the root of any order of power series

How can I calculate the root of any order of power series with Mathematica. Here I insert every quantity by hand. But I want to give a, ...
2
votes
1answer
118 views

Easier way to calculate Taylor remainder in 2nd order series

At the moment I have implemented the code for a Taylor 2nd order series for the function in three variables: $x_3^3+\frac{x_1-x_2}{x_1+x_2}$ The code builds on following expression: ...
6
votes
1answer
199 views

Trouble with shooting method for a 4th-order differential equation

I'm trying to solve the following forth-order ODE with the shooting method: $$\frac{1}{5}(y-2xy^\prime)=\frac{1}{x}\left\{\frac{xy^\prime}{y}+xy^3 \left[\frac{(xy^\prime)^\prime}{x} \right]^\prime ...
5
votes
2answers
137 views

Power series expansion

I am asking Mathematica for the first 5 terms in a power series expansion like this: ...
-1
votes
1answer
63 views

Taylor and Maclaurin series [closed]

I need to calculate limit of sin(x)*cot(tanx) at 0 using Maclaurin series I took the usual approach but did not get anything of help.
4
votes
1answer
66 views

series to rational order

The function Series requires that the expansion order is an integer, i.e. for the input ...
3
votes
1answer
70 views
-5
votes
1answer
139 views

I was wondering how to get the following result by mathematica or maple or other way [closed]

[![enter image description here][1]][1] I was wondering how to get the following result by mathematica or maple or other way I have asked this question several days ago and recently I have got some ...
0
votes
1answer
77 views

Series expansion not possible for more than two variables?

My problem is simple. I want to evaluate the series Series[(1 + x*b - x*a)/(1 + 2*x*c - 2*x*a), {a, 0, 2}, {b, 0, 2}, {c, 0, 2}]. Mathematica output looks like ...
0
votes
2answers
82 views

plotting summation to check for convergence

Our assignment was to enter in a series and check to see if it converges by graphing. However, when I attempt to graph it, I get a bunch of errors that I'm not sure what they mean. Best way to show ...
5
votes
0answers
80 views

Getting around Series[Sinc] bug

For some reason, Series expansion of Sinc around x=0 for low orders fails to generate a ...