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

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7
votes
1answer
277 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 ...
1
vote
1answer
99 views

sequence of matrix multiplications

My linear algebra book has questions that are are marked to use math software. There's probably some $50 guide on how to do everything but usually documentation has gotten me through. I've been kind ...
0
votes
1answer
130 views

Retain required terms in asymptotic expansions

I am using Mathematica 8 to do lengthy asymptotic expansions for use in statistics. In particular I have $\lambda=\beta+\epsilon+\delta+\gamma+O(n^{-5/2})$ where the first term is of order ...
6
votes
2answers
753 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 ...
1
vote
1answer
1k views

Switching the axis of a plot [duplicate]

I have a plot but the x and y axis need to be switched. The problem is that I can't explicitly solve for the other so I can change the axis. ...
2
votes
1answer
122 views

from complex function to a series

How can I express this complex function as a series? Log[ (1 - E^((I Pi (1 - a))/(b - a)) z)/ (1 - E^(-((I Pi (1 - a))/(b - a))) z) ] Where ...
11
votes
2answers
2k 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 ...
10
votes
2answers
456 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 ...
13
votes
4answers
961 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? ...
1
vote
0answers
63 views

Decimal representations of analytic values in an expansion

I have a solution to an ODE (which I call sol), which I would like to expand in terms of Log[1+x] around x=-1. I thought that: ...
8
votes
3answers
265 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: ...
2
votes
1answer
89 views

The proper way to write the input for a certain series

Mathematica tells the series below doesn't converge. I think it converges. What would the proper way to write things be as an input? ...
-1
votes
1answer
149 views

How can we suppress the asymptotic notation in Series? [closed]

Series expands a function, and also gives an idea of the asymptotic bounds of the function: Series[$\frac{1-x^3}{1-x}$] returns: $1 + x + x^2 + O(x)^5$ I'd like ...
2
votes
1answer
1k views

Laurent series expansion

Can someone share how to find a Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at zero on the annular disk $1<|z|<2$?
10
votes
1answer
304 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. ...
6
votes
0answers
109 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
1k 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 ...
1
vote
0answers
122 views

Getting poles of a Gamma functions [duplicate]

Why do the following 2 sequences give different answers? n = 1.5 Series[Gamma[0.5 - n - x], {x, 0, 2}] Series[Gamma[-1 - x], {x, 0, 2}] (..clearly the output from the second expression is ...
5
votes
0answers
108 views

Expansion of $E(i c \mid m)$ at $c\to\infty$?

Currently, I am using a Windows machine with Mathematica 8. I noticed a difference in a series expansion of the function EllipticE[] in comparison with a result ...
1
vote
1answer
81 views

Asymptotic expansion of a list

I am trying to calculate a asymptotic series expansion of a list. ...
9
votes
2answers
162 views

Unexpected behaviour of SeriesCoefficient?

Following this question/answer I discovered and played with SeriesCoefficient. In particular, I tried ...
9
votes
2answers
722 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 ...
0
votes
1answer
89 views

Formatting Equation Output Neatly

I looked around and couldn't find the answer to this anywhere, so I'm sorry if this is a bad question - I'm pretty new to mathematica. I wrote a program to help me compute some annoying series ...
4
votes
0answers
115 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: ...
4
votes
2answers
572 views

How do I solve N simultaneous equations for N variables?

I have a function: f[x_] := x + 31 x^3 + 5 x^25 Which I want to find an expansion for: ...
1
vote
1answer
144 views

Series command no longer expands arbitrary functions after aborting previous evaluation

I asked Mathematica 9 to execute the SeriesCoefficient command on a rather horrendously complicated expression. After some time I decided to abort the evaluation ...
0
votes
1answer
638 views

Asymptotic expansion, negative powers

The question was inspired by this discussion: How to expand a function into a power series with negative powers? I am interested in asymptotic behavior of a function at infinity: ...
10
votes
0answers
215 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 ...
7
votes
2answers
1k 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 ...
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, ...
15
votes
4answers
4k 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 ...
1
vote
1answer
1k views

How can I get a Taylor expansion of the Sin[x] function?

How can I get a Taylor expansion of the Sin[x] function by the power series?
2
votes
2answers
792 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
2
votes
1answer
324 views

About high dimensional integrals

I want to be able to do high dimensional integrals like, (..naively I wrote it as this..) ...
8
votes
2answers
410 views

Series expansion with irrational power

I need the series expansion of a fairly nasty function and its derivative: ...
3
votes
3answers
624 views

Limiting form of a polynomial expression

When simplifying an expression by hand, a trick that is often used is to remove terms that are lower powers of the independent variable, for instance, as $x \rightarrow \infty$, $x^2 + x$ becomes ...
0
votes
1answer
861 views

Fourier series of interpolating function result of NDSolve

I am having a tough time formulating the right question but here goes. I know that solving the pde as in here gives me an interpolating function. I understand that the interpolating function object ...
6
votes
1answer
196 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 $f=\mathcal O(x)$, $g=\mathcal O(x^2)$, $h=\mathcal O(x^3)$ ...
11
votes
2answers
671 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 ...