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

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21
votes
4answers
6k 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 ...
18
votes
4answers
543 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 ...
17
votes
1answer
1k 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 ...
17
votes
1answer
193 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
4k 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 ...
14
votes
1answer
269 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 ...
13
votes
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? ...
13
votes
1answer
468 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? ...
12
votes
3answers
514 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 ...
12
votes
0answers
255 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 ...
11
votes
2answers
783 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 ...
10
votes
2answers
529 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 ...
10
votes
2answers
182 views

Unexpected behaviour of SeriesCoefficient?

Following this question/answer I discovered and played with SeriesCoefficient. In particular, I tried ...
10
votes
1answer
259 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
1answer
274 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 ...
10
votes
1answer
144 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
1answer
346 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
0answers
408 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 ...
9
votes
2answers
842 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 ...
8
votes
3answers
298 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
454 views

Series expansion with irrational power

I need the series expansion of a fairly nasty function and its derivative: ...
8
votes
1answer
275 views

Error in infinite sum

The binary weight of the non negative integer k is defined by w[k_] := Total[IntegerDigits[k, 2]] The first values are (cf. http://oeis.org/ A000120) ...
7
votes
6answers
2k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
7
votes
2answers
601 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 ...
7
votes
2answers
108 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 ...
7
votes
2answers
2k 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 ...
7
votes
1answer
244 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}] ...
7
votes
1answer
315 views

Series expansion for small ratio of variables

I have a messy expression of variables that I would like to simplify under the assumption that certain ratios of the variables are small. For example, consider $\sqrt{x+y}\;$ expanded for small ...
6
votes
3answers
620 views

How to make all numbers equal to one in a Series?

I have many outputs of one-dimensional Series expansions for which I am only interested in the general tending with the variable. For example, I would like to be able to transform something like ...
6
votes
1answer
1k views

How to expand a function into a power series with negative powers?

Is there any way to expand this expression a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)]) (where a, ...
6
votes
2answers
229 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 + ...
6
votes
1answer
212 views

Sophistication of Series[…]

I'll give a concrete example and I hope that my general question will be clear. Say I have three variables, $f$, $g$, $h$, and I know that $f=\mathcal O(x)$, $g=\mathcal O(x^2)$, $h=\mathcal O(x^3)$ ...
6
votes
4answers
644 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: ...
6
votes
1answer
280 views

Find exponential generating function from the first few terms

The function FindGeneratingFunction computes the ordinary generating function of a sequence, given a sufficient number of initial terms. I have two questions in ...
6
votes
2answers
238 views

How to reverse irreversible function in Mathematica?

How to reverse formula $y(x)=x (\frac{1}{sin \frac{\pi x}{2}})^\alpha$ i.e. express it as $x = x(y)$ in Mathematica? I did this way ...
6
votes
1answer
95 views

Is it possible to find a function from first few terms in the expansion

Is it possible to find a function if first few terms of the expansion is known. For example if I have this series $f(x)=\frac{k^3 x^2}{6}-\frac{k^5 x^4}{120}+\frac{k^7 x^6}{5040}-\frac{k^9 ...
6
votes
1answer
131 views

Is it possible to circumvent a bug inside SeriesCoefficient?

As far as I can tell, there seems to be a bug in SeriesCoefficient: ...
6
votes
2answers
1k views

Solution of equation with power series (perturbation)

So I need to use Mathematica to find the solution of $y=x- \epsilon \sin(2y)$ as a power series in terms of $\epsilon$. I'd assume I'd need to create an equation $f=x-y- \epsilon \sin(2y)$, then ...
6
votes
1answer
168 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 ...
6
votes
1answer
103 views

`FindSequenceFunction` for sum of hypergeometric terms?

The built-in function FindSequenceFunction is quite good at recognizing hypergeometric terms, i.e. terms $c_k$ for which $c_{k+1}/c_k$ is a rational function of ...
6
votes
0answers
113 views

Design considerations behind `O` (a.k.a. BigOh, a.k.a. Landau Order)

This works without any warnings: O[Log[x]]. This raises a warning: O[x^2]. I have a few questions around this: Why is it a ...
5
votes
5answers
168 views

Isolating cross terms

I have a set of expressions that contains terms like $\frac{1}{(x + a + b)(c+d)}$. I would like to simplify the denominator so that products of $a,b,c,d$ are dropped, but $x c$ and $x d$ are kept. ...
5
votes
1answer
827 views

Asymptotic expansion

I wanted to expand a function of $x$ about $x=\infty$ and see its coefficients as a function of the parameters $m,n,q,y$ and I wrote this - but it didn't work! It gave me back a very complicated ...
5
votes
2answers
188 views

How to get the series expansion of $e^{x^a}$ at $x=0$?

I want to have a series expansion of $e^{x^a}$ or $e^{c_1x^a+c_2x^b}$ at $x=0$, but Series cannot give any useful result even if the assumption $a>0$ is ...
5
votes
2answers
102 views

Replace expression in series expansion [duplicate]

A test case: I'm trying to replace an expression inside a series expansion: Series[f[x],{x,x0,4}] ./ (x-x0)->h but it still returns ...
5
votes
2answers
129 views

Series solution of the Lane-Emden equation

I have the following problem: I'd like to show how the Lane-Emden equation would look like if someone would solve it as a series. Here's the code: ...
5
votes
1answer
433 views

Erfi[z] expansion in Mathematica: is this a bug?

Looking at the expansion: Series[Erfi[x], {x, Infinity, 1}] I obtain -I+E^x^2 (1/(Sqrt[\[Pi]] x)+O[1/x]^2) (note the ...
5
votes
2answers
124 views

Power series expansion

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

Series expansion for $\frac{x}{1- \frac{1}{x}}$

I would like to expand $\frac{x}{1- \frac{1}{x}}$ as $$\frac{x}{1- \frac{1}{x}} = x \left( 1+ \frac{1}{x} + \frac{1}{x^2} +\frac{1}{x^3} + \cdots \right) = x + 1 + \frac{1}{x} + \frac{1}{x^2} + ...
5
votes
3answers
466 views

Series coefficients for an infinite sum?

I'm working through Carl Bender's Mathematical Physics lectures on YouTube (which are great fun), and I'd like Mathematica's help solving terms in the perturbation series. It would be convenient if ...