Questions tagged [series-expansion]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
41 views

Perturbing a tensorial expression

I am new to Mathematica. I am trying to simplify an expression of the some form like: $$ n_i \sigma_{ij} n_j - \gamma n_i \hat{\sigma_{ij}} n_j = 2 + v_i x_i + \kappa E_{ij} \chi_{ji} $$ There are ...
3 votes
2 answers
203 views

Can Mathematica simplify an ordinary differential equation (ODE) by assuming a power series solution and obtain the recurrence relation?

I have an ordinary differential equation and want to solve it using power series as ψ[x_] = Sum[Subscript[a, n] x^n, {n, 0, ∞}] to obtain the recurrence relation ...
0 votes
1 answer
51 views

Truncation by coefficient size

I have a series with e.g. Chebyshevs: $\sum_i^N a_i T_i(x)$ where they are decreasing in size with increasing $i$. So now suppose I multiply two such series: $(\sum_i^N a_i T_i(x))(\sum_i^N b_i T_i(x))...
1 vote
0 answers
81 views

Strange behavior of error function series expansion at infinity

Evaluating Normal[Series[1/ (1 + xi Erf[xi]), {xi, ∞, 2}]] Normal[Series[1/( xi (1/xi + Erf[xi])), {xi, ∞, 2}]] gives ...
4 votes
1 answer
170 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: ...
2 votes
1 answer
166 views

BUG: Why is Series[] getting this expression wrong?

EDIT: Wolfram confirmed this is a bug in Series[], and they're looking into it. I'm trying to generate a 2nd-order Taylor series in theta for a complicated expression "mdel" using Series[...
1 vote
1 answer
129 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 ...
1 vote
0 answers
41 views

Identification of terms

I have the following sum on terms: ...
2 votes
0 answers
69 views

Series expansion of an action to get quadratic order terms of the perturbation

I'm trying to solve this paper (Eqs 6-11). The action is defined as $S = \frac{1}{2 \pi \alpha'}\int_{\Sigma} d\tau d\sigma(\sqrt{-\det g_{ab}}+ B_{mn} \partial X^m \partial X^n)$. where, $\tau=t$, $\...
6 votes
1 answer
233 views

Using Integrate and then Series seem to produce a wrong result

Bug introduced in 11.1.0.0 or earlier and persisting through 14.0 or later Run this: ...
1 vote
1 answer
93 views

Limit of Hypergeometric Functions

Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following ...
0 votes
0 answers
57 views

Series behavior for self-defined function

How to make Series give the correct expansion of a self-defined function? For example, for some reason, I use f[x] to represent ...
2 votes
1 answer
132 views

Asymptotic solution of a system of ODEs

I have the following system of Ordinary Differential Equations (ODEs) together with initial values ...
1 vote
1 answer
67 views

Expanding polynomials using valuation

I would like to expand the polynomial $p(\lambda) = \sum_{i=0}^{d} a_{i} \lambda^{i}$, as $F(p(\lambda), \lambda_{0})= \min_{j} [ val(a_{j}) + j \lambda_{0} ] $ with $\lambda_{0}$ being a real ...
1 vote
0 answers
117 views

How can I fully simplify sum that includes absolute value?

Consider the following sum: ...
3 votes
3 answers
181 views

Evaluating series expansion is very slow

I need in my work to get series expansion of (2 E^x x HypergeometricPFQ[{1}, {1/2 + E^-x/4, 1 + E^-x/4}, -(x^2/4)])/Gamma[E^-x/2] + x^(1 - E^-x/2) Sin[x] up to $n=...
0 votes
3 answers
181 views

Replacing function by another one each time it appears

I have expression involving Cos of some parameters. I would like to replace those Cos by their infinite series each time they do appear. I tried the following which doesn't work: ...
0 votes
1 answer
93 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 ...
9 votes
2 answers
391 views

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

Bug introduced in 12.0 or earlier, persisting through 13.2 or later Mathematica correctly identifies this sum as $\cos(x)$: ...
1 vote
2 answers
186 views

How to write code for SeriesCoefficient to work for non integral coefficients?

I have a function of $r$ which I expand at $\infty$ using Series. It is a complicated and messy function, with a parameter $0 \leq \epsilon < 1$. After expansion,...
2 votes
3 answers
153 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^...
3 votes
2 answers
171 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, ...
0 votes
0 answers
52 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 ...
0 votes
0 answers
98 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? ...
1 vote
0 answers
80 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. <...
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{...
1 vote
1 answer
106 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 ...
2 votes
1 answer
75 views

An apparent error with Chebyshev polynomials

I am on 11.0.1.0 SeriesCoefficient[ChebyshevU[n,x],{x,0,m}] returns ...
11 votes
2 answers
303 views

Is it possible to circumvent a bug inside SeriesCoefficient?

Bug introduced in 9.0.1 or earlier and fixed in 13.3. As far as I can tell, there seems to be a bug in SeriesCoefficient: ...
12 votes
2 answers
488 views

Why do I get a wrong result from SeriesCoefficient?

Bug introduced in 7.0.1 or earlier and fixed in 13.3 Consider the following code: func[x_] = Sin[x^3]/(x - 1/3); c[n_] = SeriesCoefficient[func[x], {x, 0, n}] <...
1 vote
1 answer
87 views

Proving an expression from Mathematica which is clearly visible from Plots

I have the following Mathematica code: ...
2 votes
1 answer
108 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 ...
1 vote
0 answers
90 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 ...
4 votes
1 answer
112 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 ...
0 votes
1 answer
83 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 ...
12 votes
3 answers
7k views

Laurent series expansion

Can someone share how to find the Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at $0$ on the annulus $1<|z|<2$?
0 votes
0 answers
139 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 ...
1 vote
1 answer
221 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: ...
1 vote
1 answer
50 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 ...
2 votes
0 answers
189 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 ...
1 vote
1 answer
88 views

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 ...
2 votes
4 answers
323 views

Series expansion of the integral from its numerical values

I have already asked a similar question regarding approximating Taylor series of the function from noisy data. This is another example I am having trouble with. Consider the integral $$I(x)=\int_0^1\...
0 votes
0 answers
34 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 ...
1 vote
1 answer
125 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.
1 vote
0 answers
70 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 ...
4 votes
1 answer
117 views

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

I started by defining ...
0 votes
0 answers
34 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] ...
3 votes
2 answers
257 views

Possible error with series expansion

I'm expanding the following expression around x=1 ...
1 vote
0 answers
115 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 ...
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 ...

1
2 3 4 5
18