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

learn more… | top users | synonyms (1)

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 ...
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 ...
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 ...
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: ...
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 ...
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 ...
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, ...
2
votes
2answers
86 views

Strange failure of Series and Derivative

I just spend three hours and posted two Questions trying to figure something out, and it turned out all the confusion was caused by this mysterious quirk. I want to expand g[x,v] in v at v=0, using ...
3
votes
3answers
746 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
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
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 ...
2
votes
2answers
1k views

Calculating Taylor polynomial of an implicit function given by an equation

I'd like to write a procedure that will take an equation: F(x,y,z) = 0 chosen variable: x a point: ...
3
votes
0answers
152 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
137 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
126 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
293 views

Constraint on variables in summation

Is it possible to have a constraint on variables in summation of series just similar to pattern constraint e.g. ...
1
vote
1answer
2k 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
90 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"; ...
0
votes
1answer
245 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 ...
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 ...
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
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 ...
3
votes
3answers
222 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
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? ...
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. ...
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 ...
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 ...
4
votes
2answers
680 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
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 ...
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
185 views

List manipulation - various

Update OK, Please forgive the messiness of this, but I am working with something like: ...
3
votes
2answers
150 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
836 views

About generating power series

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

Solve of cubic and quartic equations too slow

In my problem I have a third order algebraic equation for the variable sigma, all other letters are parameters. Here is it's right-hand side, left hand side is ...
1
vote
1answer
67 views

How to bound a recursive Derivative definition?

I need to convert derivatives of g[x,v] wrt v into derivatives wrt x using this relation: $$\frac{dg[x,v]}{dv}=\frac{1}{2}\frac{d^{2}g}{dx^{2}}+\frac{1}{2}\left(\frac{dg}{dx}\right)^{2}$$ This code ...
1
vote
1answer
78 views

Numerically solving a transcendental equation as a series

Given a transcendental equation $f(x,y)=0$, is there a way for Mathematica to automatically solve the equation as a series? I already know that I can use ...
1
vote
1answer
115 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
90 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
1k 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 ...
-5
votes
1answer
125 views

I was wondering how to get the following result by mathematica or maple or other way [closed]

[![enter image description here][1]][1] I was wondering how to get the following result by mathematica or maple or other way I have asked this question several days ago and recently I have got some ...