Questions tagged [series-expansion]

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

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2
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0answers
35 views

Series expansion of Lerch transcendent still buggy?

This series expansion of a Lerch transcendent seems fixed in V12. However, the following still fails: From the definition of a Lerch transcendent, ...
3
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1answer
71 views

Implement Baker-Campbell-Hausdorff expansion

I wish to calculate the recursive formula, $$\mathrm{e}^{-A_1 \Delta t/2}\mathrm{e}^{-A_0 \Delta t/2}B\mathrm{e}^{A_0 \Delta t/2}\mathrm{e}^{A_1 \Delta t/2} $$ with the BCH expansion to second order ...
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0answers
31 views

Fast way to the Taylor series expansion coefficients of multi-variable function? [duplicate]

Is there a fast method to get the coefficients of Taylor series expansion of function $f(x_1,x_2,...,x_d)$ with maximal summed partial derivative up to $n$, where $d,n$ can be relatively large (for ...
1
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0answers
39 views

Number-theoretic notation in Wolfram Cloud for iOS

I am using Wolfram Cloud Mathematica, and I want to write an equation using the mathematical symbols for Sum (like a capital sigma, but no the Greek letter) and <...
2
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1answer
57 views

Strange behaviour of infinite sum (H[n]- Series[H[n]])

Bug report filed 14.01.2020 A support case was created with the ID [CASE:4371991] EDIT It is easy to show that the workaround "limit of finite sum" proposed in the solution by user64494 leads to ...
1
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1answer
33 views

Series of implicit function (Puiseux expansion) - problem

This two one-line codes should represent the same thing, i.e. the first root of polynomial in $y$: ...
2
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1answer
43 views

Inverse of matrix up to some order

Let $A(t,s)$ be a matrix of any size (potentially large), whose entries are polynomials functions wrt $(t,s)$ of order $N$. I would like to compute the inverse $X$ of $A$ up to the order $N$ that is $...
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0answers
56 views

Smooth approximation near a non differentiable point

Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be a function differentiable for $x>0$ but non differentiable at $x=0$ (for instance $f=\sqrt{\cdot}$) and $g$ be a polynomial function. I know how to ...
2
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0answers
73 views

An asymptotic expansion for MeijerG function [migrated]

By using Mathematica Series[MeijerG[{{-(1/2)}, {}}, {{-(1/2), -(1/2), 1/2}, {}}, a x], {x,0, 0}, Assumptions -> a > 0], I got an asymptotic expansion of ...
0
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1answer
61 views

Finding ODE series solution coefficients

I am trying to solve an ODE by subbing in a series form and then looking individually at the coefficients of different powers of the variable. I'm looking at a general form of equation: $$\frac{\...
4
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0answers
76 views

How to calculate the series of this function?

I have to calculate the series of the function F[r_] := 1 - a - a*r^(5 + n)/(r^(8 + n) + 1); for r->Infinity for generic ...
1
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1answer
28 views

replacement of series expression

I want convert $$x^k\sum_{i=1}^\infty a[i]x^i$$ to $$\sum_{i=1}^\infty a[i]x^{i+k}$$ by Mathematica,since it's timing consuming and worth a little to do it by hand. from ...
1
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2answers
39 views

How to obtain the linear terms of the expansion?

Consider the following function, which is actually linear on the going-to-expand points. I want to use Mathematica to expand the function near these points, and keep only the linear terms, i.e. the ...
0
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2answers
100 views

Expand in series an equation of two variables

I am sorry if this has been asked, I don't really know how to formulate the search to find it. We have some system of equations (that are equal to zero), for example (not sure how to enter ...
3
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5answers
119 views

How do I collect different exponents together?

Suppose I have: ...
0
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1answer
47 views

Double Fourier series - value at a point [closed]

I would like to receive the value of the series at point x->1,y->1. Where is the mistake? ...
0
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0answers
32 views

Why can Mathematica detect a linear term in a series expansion only in special cases?

The series expansion Series[Sqrt[2 - 2 Cos[d k]], {k, 0, 5}] Is in odd powers of Abs[k], or more precisely ...
0
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0answers
51 views

Dealing with positive power variables in Taylor expansion and assumptions

I would like to compute the Taylor expansion of monomials whose powers are nonnegative rational variables. First, here is a function which encodes the multivariate Taylor expansion at the points <...
0
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0answers
34 views

Series expansion of explicit functions

For the following input, zv[u_, v_] := v + zv1[ u, v]/u; mvu[u, v] := D[zv[u, v], u]; Series[mvu[u, v], {u, 0, 1}] the output is ...
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0answers
29 views

Differentiation of a series with unknown coefficients

I have a function $f(x)$ expressed as $$f(x) = \sum_{k=1}^n {a_k}sin(kx)$$ and its derivative with respect to $x$ is then $$f'(x) = \sum_{k=1}^n k{a_k}cos(kx)$$ I am actually new to Mathematica and ...
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1answer
54 views

Is priority important in `Series` expansion?

I have a strange case here! The story begins with this equation ...
14
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4answers
265 views

How to convert this term to a Hypergeometric function?

term=8*(-1)^(1/4)*Sqrt[b]*q0^(3/2)*\[Kappa]* EllipticF[I*ArcSinh[((-1)^(1/4)*Sqrt[b]*r)/Sqrt[q0]], -1] This is a physical term and it is not convenient to appear ...
0
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1answer
41 views

Issue With Example of Series Inversion

I am having a problem with the Mathematica InverseSeries command. Looking at the information page here, we have the following example; ...
0
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0answers
60 views

How to print highest power of a polynomial first [duplicate]

Is there any built-in way to print the highest power of polynomial first and followed by the rest? ...
1
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1answer
131 views

Taking the power of a series gives a terribly complicated expression

I have a series $$\Phi_2=\sum_{0\leq n\leq N}a_nq^n+O(q)^{N+1}$$ whose coefficients $a_n$ are Laurent polynomials in the variables $X_1,X_2$. I want to find $\Phi^3$. This is should be pretty ...
1
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3answers
108 views

Inverting series with symbolic coefficients?

I am trying to invert the series symbolically. Is this possible in Mathematica? Example 1 - Let $p = u + au^2 + bu^3$, where $a,b$ are symbolic variables. I am trying to invert the series around $u=...
0
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0answers
38 views

Problems with the VariationalD (VariationalMethods package)

I'm facing the following problem: The VariationalD function doens't commute with the series expasion. Physically this means that if I compute the equation of motion starting with an Hamiltonian with a ...
1
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1answer
46 views

How to express the coefficients of a Series as a series itself

For example, SeriesCoefficient[ArcCos[d], {d, 0, n}] returns: ,while SeriesCoefficient[ArcCos[d]^2, {d, 0, n}] returns:
1
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2answers
60 views

Determinant of matrix with asymptotic expansion

i have determinant which each element have asymptotic expansion. $\begin{bmatrix}1+5/s+6/s^3+O[1/s^4] & 1+8/s+4/s^2+O[1/s^4]\\1+2/s+2/s^3+O[1/s^4] & 1-1/s+8/s^3+O[1/s^4]\end{bmatrix}$ ...
0
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0answers
60 views

Reduce taking long time on six equations

I have a system of equations, 4 linear and 2 quadratic, of the following form: $$L_{i,j}(x_1, \cdots, x_{12}) = \sum_{k=1}^{12} x_k c_{k,i,j}$$ for $i, j = 1, 2$, i.e. the $L_{i,j}$ are the ...
0
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1answer
73 views

Substituting and exponential term by another expression

I would like to do, in every calculation that I perform, the following substitution: $e^x -> (1+x)$. That is, every term with an exponential should become the Taylor series to the first-order ...
8
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1answer
135 views

Error Message when nothing should be evaluated

When I try to define this simple function getCoeff[SeriesData[_, _, coeff_, _, _, _]] := coeff[[2]] I get an error: ...
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0answers
29 views
2
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1answer
84 views

Series applied to an infinite sum does not work

When I have Sum[(-1)^n x^(n^2) y^n, {n, 0, ∞}] and I try evaluating ...
0
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1answer
38 views

Multivariable Taylor expansion give incorrect answer

I have a matrix $M(\text{kx$\_$},\text{ky$\_$})=\left( \begin{array}{cc} a_2 (1-\cos (\text{kx}) \cos (\text{ky}))+a_1 (1-\cos (\text{kx})) & 2 a_2 \sin (\text{kx}) \sin (\text{ky}) \\ 2 a_2 \...
4
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3answers
196 views

Eigenvectors in the limit $ \mu\rightarrow 0 $ are not the same as eigenvectors when setting $ \mu=0 $ from the beginning

I would like to find the eigenvectors of a matrix and see what the eigenvectors look like in the limit of $ \mu\rightarrow 0 $: ...
0
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1answer
51 views

Plot Taylor Polynomial

I am trying to plot Series[1/(x^2 + x), {x, 1, 5}] using Mathematica by adding this input, however its not plotting. ...
-1
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1answer
55 views

Invert a series with coefficients depending on x

Let's say I have the equation: $\qquad (8x^{15}+4x^{14}+\ldots)G^9 + (3x^{3}+\ldots)G^{8} + \ldots + (5x+4)G + x = x$. $G$ is an infinite series of $x$. I want to find the first 10 coefficients of $...
3
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2answers
57 views

Limit of an infinite summation

The above is from Maple 2019.1. Is there a way to achieve the same result from MMA12? Tried Limit[Sum[Sqrt[1 + k^2/n^3] - 1, {k, 1, n}], {n -> Infinity}] ...
0
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0answers
22 views

Fastest way to calculate Taylor expansion coefficients where the coefficients are functions of another variable that is evaluated at several poinst

I have an expression say V which is a function of 18 variables, 17 of which will be Taylor expanded to some order and the last one which will be explicitly ...
1
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2answers
95 views

What series does Mathematica use for Hypergeometric1F1?

I'm trying to get an analytical expression for Hypergeometric1F1[-a, 1/2, X] Provided a is an integer number. I tried adding ...
1
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0answers
65 views

How to replace variable in result of Series? [closed]

I am trying to replace $\text{xn}$ in $\qquad O(\epsilon )^3+\left(\text{xn}^2+2 \text{xn}\right) \epsilon +\left(-2 \text{xn}^3-4 \text{xn}^2\right) \epsilon ^2-1$ with $O(\epsilon )^2+\epsilon +1$....
1
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0answers
47 views

Expanding elliptic function of third kind

I want to expand an elliptic function of third kind in terms of $g$, where $n$, $\psi$, and $k$ are functions of $g$. For example, I defined ...
2
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1answer
63 views

Series expansion for `ProductLog[-1,x]`

Is there a way to get Mathematica to expand Series[ProductLog[-1, x], {x, -Exp[-1], 1}]? I'm on Mathematica 12, and it just returns ...
1
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1answer
78 views

Series of a product of series

I am writing a code where I have to multiply large series truncated at some order, say 100. As a result I want again a series up to order 100, and I want to get it in the fastest way. I have tried a ...
1
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2answers
96 views

Coefficient of power $p$ in polynomial expansion : Mathematica doesn't answer me

I would like to compute the coefficient in front of $n^p$ for a polynomial expansion. I wrote the following code: ...
0
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1answer
67 views

Asymptotic series : why doesn't Mathematica compute it in power of $1/x$?

Consider the following : Series[Cos[Pi/2*Sqrt[n/x]] - Cos[Pi/2*Sqrt[(n + 1)/x]], {x, Infinity, 2}] I expect mathematica to find me the $1/x$ perturbative ...
1
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0answers
23 views

Showing step-by-step answers for power series identities?

We have Sum[1/n^2, {n, 1, Infinity}] (* Evaluate to Pi^2/6 *) ..but is there a way to show how this is derived step-by-step? NOTE: I have seen step-by-step ...
3
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1answer
65 views

Multiply a Sum by a factor

A very simple question: How can I tell to Mathematica that: $\begin{equation} x*\sum_{k=0}^{\infty}\,b_kx^k=\sum_{k=0}^{\infty}\,b_kx^{k+1}\end{equation} $ I tried to multiply but Mathematica gives ...
0
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0answers
32 views

List of answer to solve with interdependant solutions

I am solving a fairly annoying matrix equation order by order in perturbation around a small parameter. I am solving for a list of free parameter that are $\beta 1[1],\beta 0[1],\alpha 0[1], \beta m1[...