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

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15
votes
4answers
5k 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 ...
13
votes
2answers
3k 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
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 ...
5
votes
4answers
440 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: ...
9
votes
2answers
761 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 ...
7
votes
2answers
492 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 ...
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, ...
3
votes
3answers
661 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 ...
7
votes
1answer
293 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 ...
3
votes
0answers
142 views

Generating function for Newton series?

The function GeneratingFunction gives generating function for Taylor series. Is there a similar function for Newton's series? $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)$$
2
votes
2answers
132 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
vote
1answer
97 views

Series expansion of InterpolatingFunction obtained from NDSolve

I am trying to obtain a series expansion of the numerical solution of a differential equation. I encounter difficulties going beyond first-order expansions which I believe might be due to my inability ...
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?
0
votes
1answer
81 views

nested series expansion

I have to invert the following matrix in which the functions U[t,x,y,r] and K[t,x,y,r] and all their derivatives are "small"; ...
11
votes
2answers
714 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 ...
6
votes
2answers
832 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
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 ...
3
votes
3answers
211 views

What is the formula for this numerical series?

I'm developing a questions game. My goal is that the score for each correct answer will increase as the user answers more questions. Initially there are 15 points for each correct answer. Every 4 ...
13
votes
4answers
999 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? ...
10
votes
1answer
318 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
1answer
91 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 ...
4
votes
1answer
384 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 ...
4
votes
2answers
608 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: ...
10
votes
1answer
188 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 ...
5
votes
1answer
117 views

Apparently contradictory output from Series and D

I defined a function as: g[b_] := Integrate[f[n]Exp[-b DA (n.u)^2], n] Obviously D[g[b], b] /. b -> 0 gives ...
4
votes
1answer
179 views

List manipulation - various

Update OK, Please forgive the messiness of this, but I am working with something like: ...
3
votes
2answers
147 views

Change all values of a series to positive values

I would like to change all values of a series to positive values. I refer to my previous question, answered in the most part, very elegantly by RunnyKine. However, there remains a slight glitch that ...
2
votes
2answers
810 views

About generating power series

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

Changing default behavior of Series to give slightly different SeriesData

I am working with complicated expressions expr that contains the symbolic function f[x] which I know to have a pole at ...
1
vote
1answer
83 views

Power series with decimal and arbitrary powers

Is there a way to make Mathematica do a series of the form: Series[ E^{\beta}$ , {x, 0, 1}] I have noticed that ...
1
vote
0answers
51 views

Best way to power series expand in multiple variables? [duplicate]

A power series expansion of a multivariate function can be performed with the Series command, which performs each expansion consecutively. The results can be a bit ...
0
votes
1answer
202 views

Multivariate series expansions to different powers

I have a function that (I believe) correctly takes the multivariate Taylor series expansion about the origin for some expression (first argument), in some variables (second argument, list), to ...
0
votes
1answer
969 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 ...