Questions tagged [series-expansion]

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

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69
votes
5answers
21k 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 \nabla)...
31
votes
4answers
13k views

Solving an ODE in power series

How do I find a series solution to an ODE? I do not mean taking the Taylor series of an exact solution; I want to solve nasty nonlinear differential equations locally via plug and chug. Surely, that ...
29
votes
2answers
5k views

How does Mathematica understand branchcuts of the complex logarithm?

Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
23
votes
2answers
617 views

Changes to Series[...] in version 11.1

Bug introduced in 11.1 or earlier and fixed in 11.2 I have updated to Mathematica 11.1 and I am shocked to see that the Series function now works differently: If ...
21
votes
1answer
400 views

Series vs Asymptotic in 12.1

The functionality of Series and Asymptotic (new in V12.1) is very similar. In fact, they are both listed in the Asymptotics ...
20
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4answers
1k 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 ...
18
votes
2answers
1k views

Series sum approximation

Since there is no closed formula, as far as I know, to find the sum of $$ \sum _{n=2}^{\infty } \frac{(-1)^n}{\sqrt{\log (n)}} $$ I used //N to find an ...
17
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4answers
3k views

Big-Oh Notation

Is there a way to have Mathematica understand Big-Oh notation? For example, I want something like: MinBigOh[...] where ...
17
votes
1answer
263 views

SeriesData sucks when it can. How do I keep SeriesData from sucking?

When I run Series[f[x]*Sin[x],{x,0,3}, Analytic->False] I get: f[x](x-x^3/3+O[x]^4) as expected. In ...
16
votes
3answers
2k views

How can I obtain an asymptotic integral expansion at infinity?

I want to find the asymptotic expansion at $x \to \infty$ of the following function: $$ I(x) \equiv \int_0^{\pi/2} e^{-xt^3 \cos(t)} dt.$$ To do this, I defined ...
16
votes
1answer
428 views

Why does $\frac{\partial}{\partial x}O\left(\left(\frac{1}{x}\right)^0\right)$ equal $O\left(\left(\frac{1}{x}\right)^0\right)$ in a series expansion?

When taking the derivative of a series expansion around a finite point, the $O(x^n)$ part is differentiated as expected. $O(x^n)$ becomes $O(x^{n-1})$ except $O(x^0)$ which stays $O(x^0)$. When ...
15
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3answers
4k 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.
15
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4answers
425 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 ...
15
votes
1answer
313 views

Why do big-O terms disappear in definite integrals since Mathematica 9?

In Mathematica 8, when I computed the following input: Integrate[Series[Cos[x], {x, 0, 2}], {x, 0, a}] Mathematica returned an expression that had a O[a^4] in ...
14
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4answers
1k views

Evaluation of a triple sum does not finish in reasonable time

I'm trying to compute the following triple sum, but no result is produced within a reasonable amount of time. What to do? ...
14
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4answers
2k views

Exponential of a Differential Operator

In Mathematica, is it possible to exponentiate a differential operator such that the operator will act on a function, $f(x,p)$? Specifically, I wondering if I can get Mathematica to do this: $\exp(c\...
14
votes
1answer
1k views

How does Mathematica find a series expansion of expressions containing logarithms when there is a singularity at the expansion point?

I am looking for a good approximation to a function containing logarithms, especially at values close to zero. When I used Mathematica's Series command I found ...
13
votes
4answers
647 views

How to find the perturbation of $x^2 − 1 = \epsilon x$?

Is there a function in Mathematica that can be used to find the perturbation solution of an equation like $x^2 − 1 = \epsilon \,x$, $x − 2 = \epsilon \cosh(x)$ or $x^2 − 1 = \epsilon\, e^x$?
13
votes
2answers
886 views

A more convenient Fourier series

Personally, I feel the design of FourierSeries/FourierSinSeries/FourierCosSeries/...
13
votes
3answers
516 views

Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
13
votes
1answer
950 views

Why is this infinite series wrongly computed by Mathematica?

Bug introduced in 7.0 and fixed in 10.0 Could you let me know if Mathematica (newer versions) is able to correctly compute this one? ...
13
votes
2answers
570 views

How to compute the residue of $e^{z-\frac{1}{z}}$ at z=0?

I've tried the following but it didn't work: Residue[Exp[z - 1/z], {z, 0}] not even this: Residue[Exp[1/z], {z, 0}] ...
13
votes
2answers
290 views

Differentiation and series expansion of dot product - inconsistent results

Bug introduced in 9.0 or earlier and persisting through 11.3 or later Bug resolved in 12.0 As of 12.0, we have an unevaluated result - inconsistent with the differentiation result, but not invalid. ...
12
votes
1answer
473 views

Dirichlet coefficients as limits: wrong

Perhaps I should have included the word "bug" in my question. Here we go There is a bug in this Limit Limit[3^s (-1 - 2^-s + Zeta[s]), s -> ∞] (* 0 *) which ...
11
votes
2answers
1k views

Series expansion of an inverse

I have to find the series expansion of the inverse function of : $\arctan\left(\frac{\ln(1+x)}{1+x}\right)$ How do I find out the series expansion of any inverse ? Note: The inverse of a function $f$...
11
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2answers
2k views

Find closed form expression for series expansion coefficients [duplicate]

Is there a built-in function that will find a general expression for the coefficient of the series expansion of a function? Series will only give the explicit ...
11
votes
2answers
2k views

InverseSeries of multiple variables and multiple equations

CONTEXT Let us consider a bit of the Universe in which we draw spheres (see a high resolution image here). Astronomers have shown that the density within these spheres could be predicted quite ...
11
votes
2answers
362 views

Why do I get a wrong result from SeriesCoefficient?

Bug introduced in 7.0.1 or earlier and persisting through 12.1 Reported to Wolfram, Inc. as CASE:3790521 Consider the following code: ...
11
votes
2answers
393 views

Mathematica 12 crashes upon taking the Log of a double series

Bug introduced in 11.1 and fixed in 12.1 I recently upgraded from Mathematica 10.4 to 12.0. Unfortunately, I am now experiencing crashes with code that was stable in 10.4. The code involves ...
11
votes
1answer
682 views

Why does Mathematica fail to series expand this simple expression?

I wanted to expand the function $(x+2)^{x+2}$ around $x = -1$, that is, using Series[(x + 2)^(x + 2), {x, -1, 2}] and Mathematica returns the same expression. ...
10
votes
2answers
417 views

How to "prepare" expression for Taylor expansion

I find myself regularly in a situation where I have an expression like $$\frac{m^2+M^2}{(m^2-M^2)^2}$$ with the assumption that $M\gg m$ and the need to expand it up to order $\mathcal{O}(M^{-2})$. By ...
10
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2answers
715 views

Series expansion with irrational power

I need the series expansion of a fairly nasty function and its derivative: ...
10
votes
4answers
5k views

How to study asymptotic behavior, built-in functions

My question is as follows. Suppose we have a function $f(r)$ and we want to study its asymptotic behavior at infinity ($r\rightarrow \infty$). For example, the function may reduce to $-\frac{a}{r}$ or ...
10
votes
1answer
936 views

Contour Integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
10
votes
2answers
263 views

Unexpected behaviour of SeriesCoefficient?

Following this question/answer I discovered and played with SeriesCoefficient. In particular, I tried ...
10
votes
1answer
218 views

How to predict the degree of the first series coefficient?

Given an expression f that is a function of x and a number x0, what is the least integer <...
10
votes
0answers
198 views

Series with ArcTan gives wrong symbolic answer in Wolfram Language

Bug introduced after 9 and persisting through 12.3.1 Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression! When calculating ...
9
votes
3answers
750 views

How do I find a series solution for $e^{-\frac{1}{2}f'(x)} \mathrm{cosh}( f(x) ) = ax + b$?

I am trying to approximate a function $f(x)$ satisfying a relation between $f(x)$ and its first derivative. How do I find a series solution for $$e^{-\frac{1}{2}f'(x)} \mathrm{cosh}( f(x) ) = ax + e^{-...
9
votes
3answers
336 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 ...
9
votes
2answers
5k 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 $1<|z|<2$?
9
votes
1answer
302 views

How could this asymptotic expansion be obtained?

I must precise that I am a very limited user of Mathematica (I can only run it from time when going at university). Working this problem, I found that $$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
9
votes
1answer
720 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 \...
9
votes
0answers
131 views

Bug in SumConvergence

Bug introduced in 10.0.1 and fixed in 12.0.0 Version 11.2.0.0 on MacBook Pro: ...
8
votes
2answers
2k views

Finding the coefficient of a certain power in a generating function

I want to compute $(t^1+\dots +t^5)(t^2+\dots+ t^6)(t^3+\dots +t^9)$ and find the coefficient of $t^{15}$ for example. $t$ is an indeterminate Now in Maple this is simply as ...
8
votes
3answers
535 views

Declaration of abstract matrices to perform series expansion on them

I would like to have abstract matrices M and S to get out the coefficients of matrix power series however it treats M and S as numbers even if i checked that M.S - S.M != 0. I attach my code below: <...
8
votes
2answers
447 views

Generating terms of the Stirling series

The Stirling series starts as follows: $$n!=\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n} \left\{1+\frac{1}{12n}+\frac{1}{288n^{2}}-\frac{139}{51840 n^{3}}-\frac{571}{2888380 n^{4}}+O\left(n^{-5}\right)\...
8
votes
3answers
469 views

Series expansion wrong

I had to work with some series expansions lately, and at some point I realised that something was becoming inconsistent at some point. It seems that applying Factor ...
8
votes
2answers
288 views

How to represent $f(x) = (y-x)^k \log(y-x)$ as a summation of the form $f(x) = \sum\limits_{j=0}^\infty \cdots$?

I am having a lot of trouble working with summations in Mathematica, and this is unfortunate as it is my main use case for the application My latest summation issue is the following. I am trying to ...
8
votes
2answers
1k views

Using Fourier Series to acquire Nonlinear ODE Periodic Solutions

For the following Cauchy problem of a non linear ODE: \begin{equation} \ddot{x}=-|x|^{1/3} \end{equation} which satisfies the initial conditions $x(0)=1, \dot{x}(0)=0$, I am aware that there are ...
8
votes
2answers
733 views

Analytical approximation of indefinite integral on a given interval to a given precision

I'm looking for an analytical approximation of $\int_a^b e^{-x^2}\mathrm{erf}(x+c) dx$ that would be accurate to precision $\varepsilon$ for $a,b,c$ within a certain range. How do I ask Mathematica ...

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