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

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2
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0answers
24 views

Differentiation and series expansion of dot product - inconsistent results

If I differentiate a dot product, I get the result I expect D[a.b[x], x] (* a.b'[x] *) However, a series expansion of the same expression gives a very different ...
4
votes
1answer
107 views

Series expansion of $(x;x)_\infty$ at $x=1^-$?

The so called Euler function is implemented in Mathematica as QPochhammer[x, x] I have been trying to obtain its leading behavior for $x\to 1$ from below. ...
1
vote
1answer
55 views

How can I make Mathematica list the terms in this series?

I am using perturbation theory to solve a problem of the following form: $$ R(h,\theta)f(\theta) = h g(\theta) = 0 $$ where $h$ is small, and I assume $$ \theta = \sum_{i=0}^\infty \theta_i h^i $$ ...
0
votes
0answers
5 views

How to derive a Taylor series from the ones we know (cosx, sinx, …) [migrated]

If we know the Taylor expansion for the cos(x) function around 0, how can we use it to derive the Taylor expansion of a similar function (cos(x+π/4)) around 0? I do know how to get the Taylor series ...
1
vote
1answer
62 views

Automation of Perturbation Solution

I'm trying to solve an equation of the form $$ R(\theta)f(\theta)+hg(\theta) = 0 $$ for small $h$, where $R$, $f$, and $g$ are functions. I've assumed a power series expansion for $\theta$ in terms ...
1
vote
0answers
41 views

Behavior of solution to a ODE near a singular point [closed]

I want to see how the solution to the following ODE looks near z=z0. ...
3
votes
0answers
40 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
172 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
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. ...
2
votes
0answers
74 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
86 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
75 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) - ...
3
votes
1answer
27 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
63 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 ...
0
votes
2answers
99 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
59 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
100 views
1
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1answer
115 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
143 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: $$ (A+B)^{-1}=A^{-1}-A^{-1}BA^{-1}+A^{-1}BA^{-1}BA^{-1}-...
8
votes
3answers
183 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
51 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
89 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 ...
4
votes
1answer
152 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? <...
0
votes
1answer
57 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 $$ G(...
1
vote
2answers
133 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, ...
2
votes
0answers
33 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
64 views

How to expand an infinite product?

How can I use Mathematica to expand such a product (only need a finite number of terms): $$\prod^{\infty}_{n=1}\frac{({1-yq^{n+1}})({1-y^{-1}q^n})}{(1-q^n)^2}$$
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 ...
1
vote
1answer
35 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, ...
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 a_2}+\...
2
votes
1answer
121 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: ...
5
votes
2answers
138 views

Power series expansion

I am asking Mathematica for the first 5 terms in a power series expansion like this: ...
-1
votes
1answer
64 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.
3
votes
1answer
70 views

SolveAlways 'never' completes

After allowing Mathematica to run on ...
5
votes
5answers
165 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 ...
6
votes
1answer
208 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 \...
10
votes
3answers
422 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.
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 ...
-5
votes
1answer
140 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
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 ...
6
votes
0answers
82 views

Getting around Series[Sinc] bug

For some reason, Series expansion of Sinc around x=0 for low orders fails to generate a <...
1
vote
3answers
114 views

How to find the leading term in the expansion for large value of the parameter

My question is related to one I already asked (previous question). The equation I am working with is cumbersome: ...
4
votes
1answer
67 views

series to rational order

The function Series requires that the expansion order is an integer, i.e. for the input ...
1
vote
1answer
98 views

Problem with complex conjugation of series data in mathematica 10

ComplexExpand in Mathematica 10 doesn't work with series data: Taylor = Series[Exp[I x], {x, 0, 4}]; ComplexExpand[Taylor\[Conjugate]] gives 1+I x-x^2/2-(I x^...