Questions tagged [series-expansion]

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

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Why does Mathematica get the infinite sum of this representation of the taylor series of $\cos^2(x)$ wrong?

Mathematica correctly identifies this sum as $\cos(x)$: Sum[((-1)^n x^(2 n))/(2 n)!, {n, 0, Infinity}] Mathematica also correctly identifies this product of sums ...
Samuel Martineau's user avatar
2 votes
3 answers
140 views

Find Generalized Series with Symbolic Variable

CoefficientList[Series[Exp[x], {x, a, 3}], x] Gives the following expression, $$ \left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
Torkoal's user avatar
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3 votes
2 answers
161 views

First argument -h is not a valid variable

I am trying to simpligy my life by defining a function that gets the series expansion of $f(x+h)$ and $f(x-h)$ (at $x=0$) by the following code, ...
kichapps's user avatar
0 votes
0 answers
46 views

Limit giving indeterminate result

I have a function $r_h(v)$ given by, $$r_h (v) = \left(\frac{m_0}{2 g} e^{-g v_f} \left(-1+e^{-g(v - v_f)} \right) \right)^{1/5}$$ where $m_0$ and $g$ are just numbers. I want to take the limits of ...
mathemania's user avatar
0 votes
0 answers
96 views

How to solve recurrence equation using RSolve?

How can I solve the following recurrence equation while I dont have the initial values? Is it possible to solve this using RSolve? ...
mehrosadat ebrahimi's user avatar
1 vote
0 answers
78 views

Series expansion message with special functions

I am trying to expand the following expression in alp,eps. Since the expression involves Hypergeometric2F1, I am using HypExp. <...
BabaYaga's user avatar
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1 vote
0 answers
25 views

Finding the coefficients of a decomposition of complicated expression into products of special functions

I have the following expression which arises while studying minimal models in CFT. The 4-point amplitude of scalars is given by $$G(z,\overline{z}) = ((1-z)(1-\overline{z}))^{\frac{m-1}{m+1}}\mathcal{...
QFTheorist's user avatar
2 votes
1 answer
73 views

An apparent error with Chebyshev polynomials

I am on 11.0.1.0 SeriesCoefficient[ChebyshevU[n,x],{x,0,m}] returns ...
მამუკა ჯიბლაძე's user avatar
1 vote
1 answer
82 views

Proving an expression from Mathematica which is clearly visible from Plots

I have the following Mathematica code: ...
codebpr's user avatar
  • 899
1 vote
0 answers
85 views

Trying to use Linear Optimization to solve inequalities

I am trying to use Mathematica to get bounds on some quantities using a certain procedure which is called conformal bootstrap. For the users who don't come with a background in HEP, I am basically ...
QFTheorist's user avatar
2 votes
1 answer
102 views

How to convert DifferenceRoot into a special function?

Clear["Global`*"]; f[z_] := z^(2 m) /(1+z)^m res = SeriesCoefficient[f[z], {z, -1, -1}, Assumptions -> Element[m, PositiveIntegers]] The result ...
lotus2019's user avatar
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1 vote
1 answer
100 views

Series expansion of Beta function in Mathematica

How one should do the series expansion of Beta function $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ around an arbitrary negative integer let's say $x=-k$ and $y=-l$? I want a symbolic expression ...
Hkw's user avatar
  • 39
1 vote
1 answer
108 views

A request on the kernel of the function ”SpheroidalEigenvalue“ in Mathemitica 11

I want to understand the algorithm of the function "SpheroidalEigenvlaue" in Mathematica 11, especially the code of the following command. Then I will try to use this algorithm to reproduce ...
amon xu's user avatar
  • 31
4 votes
1 answer
110 views

A simple series expansion which seems to be wrong

Trying to answer this question, I made the following input FullSimplify[SeriesCoefficient[ArcTanh[x]/(1 + x^2), {x, 0, n}]] I shall not type the results but, not ...
Claude Leibovici's user avatar
0 votes
1 answer
71 views

Series expansion for expression with parameter?

I would like to compute the following expansion. Series[(A + p/x^a)^2, {x, 0, 1}] where $a>0$. However Mathematica simply returns the expression, unless I ...
korni1990's user avatar
  • 307
0 votes
0 answers
115 views

How to solve or test the interval of Uniform Convergence of function series?

How to solve or test the interval of Uniform Convergence of function series? (ref2) e.g. $\sum_{k=1}^{\infty}\left(z^{k-1}-z^k\right)$ The convergence interval of this series can be got by ...
lotus2019's user avatar
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1 vote
1 answer
215 views

Discrepancy with Hurwitz Zeta function

I've come across an issue while using Wolfram Mathematica that I don't quite understand. Consider the following symbolic computation: ...
stefan_chem's user avatar
1 vote
1 answer
45 views

How to obtain a list of pairs of exponents in a double series expansion?

Let's say we have a function of two variables $f(x,y)$ and we work out its Taylor expansion up to some power. I would like to use Mathematica to construct a list of all exponents that appear in the ...
user1620696's user avatar
2 votes
0 answers
185 views

How to approximate an exponential series?

Consider the following expression $$ y_j= \sum_{k=0}^{L} \frac{e^{-\sum_{i=-k}^k(k-|i|)x_{j+i}}-e^{-\sum_{i=-k}^k(k+1-|i|)x_{j+i}}}{\sum_{i=-k}^k x_{j+i}}\tag{1} $$ for $1\leq j \leq L$. Given smooth ...
sam wolfe's user avatar
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0 votes
1 answer
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Comparing two power series and extracting their coefficients

I have a standard mathematical problem that I was wondering how to solve efficiently by mathematica. Here is the problem. I have two power series expansions of a function ...
Physics Moron's user avatar
0 votes
0 answers
32 views

Weird expression for function Series-Expansion with Gamma function for different values of gamma coefficient

I extract the function jin[r] by solving eqsynin, and then I develop the function's series (around zero) to generate an equation for m1in and m2in based on esyn and gamma, knowing that the function ...
Pantelis Ashikkis's user avatar
1 vote
1 answer
117 views

How to expand $\frac{1}{(1-z-z^2)}$ into a power series [closed]

How can I get Mathematica to expand $$\frac{1}{(1-z-z^2)}$$ into a power series so that I can pick out the coefficients.
Vectorizer's user avatar
1 vote
0 answers
66 views

How to accelerate Inverse[] for positive definite matrices symbolically?

I am trying to construct a positive definite matrix based on the multiquadric radial basis function (RBF) for a set of thirteen points symbolically in order to later approximate the Laplacian operator ...
Fazlollah's user avatar
  • 1,843
4 votes
1 answer
116 views

Why `AsymptoticSolve` doesn't work for a multivariate implicit function?

I started by defining ...
Nekomiya Kasane's user avatar
0 votes
0 answers
26 views

Attempt to evaluate a series returning Indeterminate while running Plot

I have a function in the form of a Series from a prior calculation: sol2D = SeriesData[a, 0, {Rational[3, 16] Pi, Rational[-5, 2], Rational[3, 4] Pi}, 0, 3, 2] ...
CuriousOkapi's user avatar
1 vote
0 answers
95 views

Taylor series loop

I'm a beginner not only in Mathematica but also in programming in general, and so I'm not really sure where my problem lies exactly and I'd be glad to receive any guidance. Using the Taylor series for ...
milf_and_cookies's user avatar
2 votes
1 answer
81 views

Why does Series give two different results for given function?

I want to do Series of this function x^12 Cos[y x]^2 Sin[y x]^6 Sin[(-1 + Sqrt[3]) y x]^2 around ...
Martha97's user avatar
  • 349
2 votes
1 answer
107 views

Making Series Solutions Look Nicer

My students and I are using AsympototicDSolveValue[] to find power series solutions to linear differential equations at 0. For example, the following code gives me a solution up to degree 7. I'm ...
B flat's user avatar
  • 5,409
0 votes
1 answer
119 views

Can Mathematica estimate this complex function?

Mathematica has given me a function in $x,r$ given by ...
Matthew Neil's user avatar
2 votes
1 answer
145 views

How to obtain the Taylor expansion of any function? [duplicate]

How to obtain the Taylor expansion of any function? Like the Taylor expansion of any function in the picture. How can I obtain the Taylor expansion of any function if I input it? ...
csn899's user avatar
  • 2,879
0 votes
0 answers
23 views

Different results for same output

In a power series solution method, I am trying to find the roots of an equation. By changing the parameters I need to get the roots. The problem is the parameter value is provided as 0.15 which gives ...
supragyan priyadarshinee's user avatar
4 votes
4 answers
209 views

Solving PDE with power series

I would like to solve the PDE $$\partial_{x}f(x,y) + f(x,y)^2 = g(x,y)$$ with $f(0,0)=0$ and $\partial_y f(0,0)=0$ using a power series ansatz, i.e. I have an explicit expression for $g(x,y)=\sin(x+y)\...
António Borges Santos's user avatar
0 votes
0 answers
69 views

Approximating Exp[-x] in partial fraction form [duplicate]

I'm looking to obtain order-$k$ approximation of $\exp(-z)$ for real-valued $z$. $$R_k(z)\approx \exp(-z)$$ The constraint is that I need the result in partial fraction form, ie: $$ \begin{equation} ...
Yaroslav Bulatov's user avatar
3 votes
1 answer
64 views

Approximating exponential generating function (EGF) from values of generating function (OGF)

I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
Yaroslav Bulatov's user avatar
4 votes
1 answer
335 views

Zassenhaus formula in Mathematica

I'm looking for Mathematica implementation of Zassenhaus formula -- given two matrices $A,B$, truncate the expansion of $\exp(A+B)$ from this paper: $$e^{t(A+B)}= e^{tA}e^{tB}\prod_{n=2}^\infty e^{t^n ...
Yaroslav Bulatov's user avatar
2 votes
2 answers
208 views

Series expansion using binomial theorem in Mathematica

The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by $$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
VH84's user avatar
  • 179
0 votes
1 answer
46 views

Solving series solution of differential equation

AsymptoticDSolveValue[2x*y''[x] -(3+2x)*y[x] +1 == 0, y[x], {x, 0, 5}], this differential equation command, is not outputting the correct solution. The solution should be like y=C1(1+(1/3)x-(1/6)x^2-(...
haha97894's user avatar
0 votes
0 answers
42 views

Solving differential equation to series solution

I tried to solve y''+x^2y=0 this differential equation to the series solution, so I put the command of AsymptoticDSolveValue[y''[x]+x^2*y[x]==0, y[x], {x,0,5}] like this, but the output shows like ...
haha97894's user avatar
0 votes
1 answer
82 views

Expansion of standard inverse normal cdf

Let $\Phi: x\rightarrow y$ be the CDF of a standard normal distribution. The range of $\Phi(x)$ is $[0,1]$ and the domain is all real numbers. I want to get a series expansion of $\Phi^{-1}(y)$ around ...
user1936752's user avatar
1 vote
2 answers
90 views

Series from an integral and output as a function

I have a simple question, I am just stuck on syntax. I want to have a series of function $Z(\lambda)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} d x e^{-\frac{x^2}{2 !}-\frac{\lambda}{4!} x^4}$ ...
Хранитель Рощи's user avatar
4 votes
3 answers
108 views

Why earlier terms generated from AsymptoticDSolveValue change when increasing the order?

I was comparing my hand solution with Mathematica's. I noticed the particular solution terms generated for a Frobenius series solution change depending on the order. This should not happen. As when ...
Nasser's user avatar
  • 137k
2 votes
1 answer
67 views

Then can the result of the general term formula be written in subsection form?

s[n_] = n^2 - 2 n + 3 RSolve[a[n + 1] == s[n + 1] - s[n], a[n], n] The above example shows that the general term formula of the sequence should be in a piecewise ...
csn899's user avatar
  • 2,879
1 vote
0 answers
75 views

SeriesCoefficient stops working on EllipticTheta in v13.2

In v12, the following SeriesCoefficient computation gives the expected result, ...
Lelouch's user avatar
  • 513
4 votes
0 answers
134 views

InverseSeries giving incorrect result

Somehow in Mathematica 13.2.0.0, InverSeries generates incorrect results. Let's look at the following two series that differs from each other by a constant number &...
mastrok's user avatar
  • 591
4 votes
2 answers
131 views

Looking for the asymptotics of an asymptotics

I am trying to polish my second answer to this question in Mathematics Stack Exchange. The problem is to find the asymptotics of $t$, solution of the implicit equation $$\color{blue}{\left(1-2 x^2\...
Claude Leibovici's user avatar
1 vote
1 answer
134 views

Speed up a infinity series

Is there any trick to speed up the plotting of my function u[x,t]? ...
Lavender's user avatar
  • 119
3 votes
2 answers
123 views

Is there something faster than AsymptoticDSolveValue for getting the terms in a power series solution?

Here's an example of a differential equation which Mathematica 13.1 just returns without solving ...
Mr. G's user avatar
  • 335
4 votes
2 answers
143 views

Mathematica flips the sign of a Maclaurin series

I have the following Mathematica code: ...
codebpr's user avatar
  • 899
0 votes
0 answers
88 views

Recursion error for a series expansion while using RGTC code

I am trying to use the RGTC code found on this website to calculate the series expansion for the given differential equation. I get a recursion error when I use this code: Recursion depth of 256 ...
codebpr's user avatar
  • 899
1 vote
2 answers
136 views

Finding an elementary function growing asymptotically as the integral of a sequential product

I am trying to understand how grows the function $f:\mathbb{N}\to\mathbb{R}$ Integrate[(1-Product[1-Exp[-2*j*t/(k*(k+1))], {j, 1, k}]), {t,0,\[Infinity]}] for $k\to\...
Penelope Benenati's user avatar

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