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

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0
votes
1answer
33 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
36 views

Implementation of Series function

I'm trying to evaluate the partial derivatives of some function F[x,y] at some point, i.e. ...
2
votes
2answers
697 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
0
votes
0answers
23 views

What is the common term of this series? [closed]

Consider the following series of numbers . 3,5,17,257,65537 What is the common term of this series? (n is a positive integer)
0
votes
1answer
84 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
61 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
189 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
36 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: ...
1
vote
1answer
84 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 ...
2
votes
1answer
131 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
80 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 ...
4
votes
1answer
86 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
1answer
53 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
2answers
85 views

eccentric anomaly expansion equation

I try to use this simple algorithm (paper) to calculate the Eccentric Anomaly expansion: ...
0
votes
0answers
27 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
60 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
80 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
53 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
95 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
75 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
89 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: ...
3
votes
4answers
154 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
67 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
50 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
99 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
48 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
62 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
76 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
55 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
102 views

Bessel derivatives

I tried to expand BesselJ[k,x] function into a Taylor series with Series command. Here both ...
0
votes
1answer
94 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
80 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
111 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
115 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
132 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
79 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 ...
0
votes
1answer
85 views

How could I expand this equation

How could I expand this equation (0.837 + x^0.5 + x)^4.870 Neither Expand[(0.837 + x^0.5 + x)^4.870] nor ...
0
votes
1answer
58 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
98 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
2answers
57 views

Expand expression into quotient of negative power series

I have a function T(z) and want to expand it to the following form: At the end I want to have a list of equations, which can calculate a0, a1, a2, b0, b1 and b2 with different values of h1, h2, ...
1
vote
2answers
142 views

Making mathematica do regulated integrals

Consider the integration, $\int _0 ^\infty dx\ x \tanh( \pi x) \sqrt {x^2 + a^2 } $ where $a$ is a real number. This integral is divergent. We note that an asymptotic expansion of the integrand ...
1
vote
1answer
66 views

Reducing a multi-variate polynomial to have terms upto certain degree

I have a polynomial in x and y. I want to keep all the terms with (combined) degree 4. Any pointers will be appreciated. Simple tricks for doing the same with univariate polynomial don't seem to work. ...
2
votes
1answer
104 views

Symbolically Expanding One-Variable q-hypergeometric Series

In general, how do I get Mathematica to symbolically expand various infinite series to an arbitrary degree of accuracy? I deal with q-hypergeometric series quite frequently, and specifically want to ...
9
votes
1answer
538 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 ...
4
votes
1answer
200 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 ...
0
votes
1answer
95 views

Evaluate integral of a series

I want to evaluate this integral, but it won't work. Does anyone knows why? ...
5
votes
2answers
154 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 ...
3
votes
3answers
167 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 ...