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

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2
votes
3answers
128 views

Extracting coefficients from a power series

I have a function defined explicitly as a power series: $$\sum_{n=0}^\infty{T_n}\frac{x^n}{n!}=\frac{\frac{x^3}{3!}}{e^x-1-x-\frac{x^2}{2!}}$$ and I would like to extract the coefficients $T_k$ as ...
0
votes
0answers
39 views

How to make an effective use of Mathematica to evaluate integrals, series, recognize closed forms, and do other manipulations [on hold]

I would like to know if there are resources to learn how to use the functions of Mathematica in the most effective way in order to evaluate difficult integrals and series or find possible closed forms ...
2
votes
1answer
32 views

Truncating power series

I have the following typed up so as to truncate higher powers of $\tau$, ...
1
vote
1answer
39 views

Simplify a series expansion including product and multiplication

I have a following expression $$ f=\Pi_{i=1}^n \left[ 1 + \frac{1}{t} +\frac{ (m+i)}{t^2} +\frac{ (m+i)^2}{t^3} + \cdots \right] $$ here $m,n,t$ are positive integers. I want to obtain a series ...
1
vote
1answer
41 views

Series expansion for rational function with weird powers of variable?

Consider the following series expression: Series[1/(a+b x^(1/3)+c x^(4/3)),{x,0,1}] The result comes out appropriately: 1/a - b x^(1/3)/a^2 +b^2 x^(2/3)/a^3 ...
1
vote
0answers
78 views

Asymptotics: replacing all arguments in a trig function with dummy variable after manipulation

I'm doing manipulations on functions of the form $$\phi(x,z,t) = \sum_n a_n(\epsilon x,\epsilon t,z)e^{in( k_ox -\omega_o t + \epsilon \theta(\epsilon x,\epsilon t)))} e^{(k_o+\epsilon k(\epsilon ...
1
vote
1answer
103 views

divergent part of a series

I am trying to find a function that gives me the divergent part of series in any case. The method "Series[]" does not work because Series[42,{n,0,-1}] gives as ...
0
votes
1answer
39 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 ...
0
votes
1answer
41 views

Series expanding an expression to an arbitrary power

How can I get Mathematica to directly produce the series expansion of an expression such as $(1+1/x)^n$, where $n$ is an arbitrary positive integer? Note that it's important in my expression to keep ...
2
votes
2answers
718 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
1
vote
1answer
95 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 ...
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 ...
2
votes
2answers
141 views

How to get the series coefficient of a function defined by a differential equation [duplicate]

I need a Taylor series approximation for $x(t)$, which is defined by the following differential equation. $\frac{dx}{dt}=-k_1 x+(1-x)k_2 e^{-k_3 t}$, $x(0)=0$ ...
3
votes
0answers
116 views

Generating function for Newton series?

The function GeneratingFunction gives generating function for Taylor series. Is there a similar function for Newton's series? $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)$$
7
votes
2answers
264 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 ...
1
vote
1answer
67 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 ...
0
votes
1answer
48 views

Speeding up ReplaceRepeated while truncating to desired order

I need to program in an algorithm that recursively makes algebraic replacements which leads to an utterly complicated algebraic function of $x$, but whose final result is only needed at fixed order in ...
2
votes
0answers
47 views

Implementation of Series function

I'm trying to evaluate the partial derivatives of some function F[x,y] at some point, i.e. ...
0
votes
1answer
93 views

Elegant way to do series expansion around x+$\epsilon$

Suppose I have a function $p(x+\epsilon,y-\epsilon,t)$, and I want to expand it around $(x,y)$ like $$ p(x+\epsilon,y-\epsilon,t)=p(x,y,t)+\dfrac{\partial p}{\partial x}\epsilon+\dfrac{\partial ...
0
votes
0answers
73 views

Best way to determine polynomial coefficients in series expansion

I would like to solve the equation $$h'(\boldsymbol{x}_1)\left[B_1\boldsymbol{x}_1+g_1(\boldsymbol{x_1},h(\boldsymbol{x}_1))\right]=B_2h(\boldsymbol{x}_1)+g_2(\boldsymbol{x}_1,h(\boldsymbol{x}_1))$$ ...
8
votes
1answer
199 views

Why is this infinite series wrongly computed by Mathematica?

Could you let me know if Mathematica (newer versions) is able to correctly compute this one? Sum[(-1)^(n + 1) Cos[3^n x]^3/3^n, {n, 1, Infinity}]
0
votes
1answer
41 views

Series not evaluating to numerical answer [closed]

I am working on an example from a book (An Engineer's Guide to Mathematica). The following code is in the book: ...
3
votes
1answer
140 views

Is there a function that, given a fraction, will return the general term of its infinite series expansion?

Is here some way to expand a fraction to an infinite sum in mathematica, i.e., a series? I want the general term of the series. For example, $\frac{2}{3(x-1)^3}$
0
votes
1answer
82 views

Series expansion for $\frac{1}{x+1}$ in terms of $\frac{1}{x-1}$

I would like to expand a function as $$\frac{1}{x+1} = \frac{1}{x-1+2} = \frac{1}{x-1} \frac{1}{ 1+\frac{2}{x-1}} = \frac{1}{x-1} \left[ 1- \frac{2}{x-1} + \left(\frac{2}{x-1}\right)^2 + \cdots ...
5
votes
1answer
92 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} + ...
1
vote
2answers
88 views

eccentric anomaly expansion equation

I try to use this simple algorithm (paper) to calculate the Eccentric Anomaly expansion: ...
0
votes
0answers
30 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
1
vote
0answers
61 views

Why does Mathematica not go to specified order in series?

For some reason Mathematica will not evaluate this asymptotic series to the requested order. Inputting: ...
0
votes
1answer
94 views

How to power-series expand determinants?

Say $g$ is a matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and ...
-1
votes
1answer
61 views

Plot CPU time vs iteration? [duplicate]

I would like to generate a plot of CPU time vs number of iterations. For example, if I were to calculate the solution of a system of differential equations in state-space form using the summation of ...
0
votes
1answer
103 views

How to calculate the unknown quantity in an infinite series?

I'd like to calculate x value in this equation. Basically, I tried to 2 types of method which are FindRoot and NSolve. But, I have failed the calculation caused by these errors up to now. If ...
0
votes
1answer
81 views

Problem of how to manipulate Taylor/McLaurin series

I'm new to Mathematica so I don't know very well the program itself. I would like to manipulate as he does this person in youtube: http://www.youtube.com/watch?v=fCJHvQaGNiQ But I put all the ...
0
votes
3answers
96 views

How to use Assumptions in a Series Expansion

I want to series expansion the expression $\frac{1}{2} \left(e_1+e_2-\sqrt{e_1^2-2e_1e_2+e_2^2+4V_{12}^2} \right)$ up to second order in $V_{12}$ using the assumption $e_1>e_2$. So I tried ...
0
votes
0answers
28 views

Speed up series expansion [duplicate]

I have a rather complicated function that I need to expand it in series, which is quite straightforward but takes a long time. I was wondering if there is a way to speed up such a calculation: ...
4
votes
4answers
183 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: ...
0
votes
1answer
75 views

How do I plot a Taylor polynomial of various degrees in two variables? [closed]

I want to plot the Taylor polynomials of $f(x,y)= Sin(1 + x + y^2)/(4 + x^2 + y^2)$ of degrees 4 and 7 around the point $(0,0)$ over the rectangle $[-\pi,\pi]\times [-\pi,\pi]$ I am currently using ...
0
votes
3answers
88 views

Create polynomials from Series

This question actually doesn't have quite a lot to do with the Series function, but I don't know how to describe my problem. So here's the thing. I'm trying to ...
1
vote
1answer
52 views

Series and that old ivar error

I am trying to compare plots of the function (1+x)^{1/3} and its power series. The code for the power series generates an error message referring to an invalid variable: 'General::ivar: -1.09993 is ...
2
votes
1answer
103 views

Remove singularity at zero from $-((i^{-n} (-1 + i^n)^2)/(n^2 π^2))$

The expression h[n_] = -((I^-n (-1 + I^n)^2)/(n^2 π^2)) is real for all integers n. Although indeterminate at ...
0
votes
1answer
53 views

Plot Series output? [closed]

I tried to search if similar questions existed but I have not found anything. Can someone help me with this code? ...
0
votes
1answer
70 views

Working with means and variances

My goal is to work out on the following equation: $$Mean(P)(1-Mean (P))\left (h + (1 - 2 h) (1 - \frac {Var (P)} {Mean (P) (1 - Mean (P))}) (1 - Mean (P)) + \frac {Var (P)} {Mean (P) (1 - Mean ...
2
votes
1answer
97 views

Taylor expansion of function of vectors—simplifying the output

I've got a real-valued function of several vectors $f(u,v,w)$ formed by taking scalar products of linear combinations of the vectors, I want to Taylor expand around small $v$ by writing $$f(u,\delta ...
1
vote
1answer
58 views

Compose two special power series expansions

I have two functions $A(x), B(x)$, given in a special power series form: $A(x)=1-x^{2}\left(\frac{a}{10}-\sum_{k=1}^{9}b_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$ ...
2
votes
2answers
116 views

Bessel derivatives

I tried to expand BesselJ[k,x] function into a Taylor series with Series command. Here both ...
1
vote
1answer
96 views

About doing an sum

I want to explain this summation to Mathematica, For $a >0$ define $n_u, n_d \in \mathbb{Z}$ such that if $\{ a \} \leq 0.5$ then $n_u = [a]-1$ and $n_d = - [a]$ and if $\{ a \} > 0.5$ then ...
0
votes
0answers
100 views

A power series expansion

Consider the function, $f(z) = z tanh(\pi z) log (z^2 + a^2)$ for some $a>0$. Now I am considering 3 different situations, $z = i(n+0.5) - i\epsilon + \delta - it$ for $n \in \mathbb{Z}$ and ...
6
votes
1answer
112 views

Is it possible to circumvent a bug inside SeriesCoefficient?

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

Matrix exponential MatrixExp[] vs Sum[MatrixPower[]] doesn't match?

I might be an idiot, but I cannot get the manual expansion of e^At to match the MatrixExp[A t] result. For example, I have the following: ...
0
votes
1answer
163 views

How to simplify power series in terms of functions

For example, after some computations mathematica outputs $$ \sum_{k=0}^{\infty} \frac{k g_k z^{k+4}}{(1-z)^3} $$ and we assume that $$G(z)= \sum_{k=0}^{\infty} g_k z^k $$ so the question is: how ...
1
vote
1answer
92 views

How to expand function $\cos(y+i\log{x})$ in powers of $x$?

I have the following, probably very simple question. How can I get $\it{Mathematica}$ to power expand function $\cos(y+i \log(x))$ in powers of $x$? This function obeys a well defined Laurent ...