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

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5
votes
2answers
178 views

Series solution of the Lane-Emden equation

I have the following problem: I'd like to show how the Lane-Emden equation would look like if someone would solve it as a series. Here's the code: ...
2
votes
0answers
60 views

Changing variables in a series expansion

I want to compute the Taylor expansion of some pure function f[x_], but then perform a change of variables in the resulting expression. So, for example, the output ...
1
vote
1answer
40 views

System of equations from coefficients of series

I have two functions $F_1(a,b,x)$ and $F_2(a,b,x)$ I expand these functions about the point $x=0$ Series[F1,{x,0,2}] Series[F2,{x,0,2}] the coefficients are ...
2
votes
3answers
73 views

Series: two varibles, how to eliminate the (product) higher order

I have a function of $f(x,y)$, where $x,y$ are very small numbers. I want to series expand it to the $3$rd power. However I don't want the terms such that $x^2y^2$, $xy^3$ etc. because I would reckon ...
1
vote
2answers
60 views

A series expansion

It might be a silly question. Actually I'm facing a problem expanding $(1+\frac{2}{x})^{\frac{i}{2}}$ for small $x$. Mathematica can not expand it. But it can of course expand ...
10
votes
1answer
146 views

How to predict the degree of the first series coefficient?

Given an expression f that is a function of x and a number x0, what is the least integer ...
2
votes
2answers
88 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 ...
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
66 views

How to use PadeApproximant with a defined Derivative?

I need to generate a Pade approximant expansion in v of the following: $$g(x,v)\equiv ...
1
vote
1answer
49 views

Failure of Series[] for hypergeometric functions

I am encountering peculiar errors when asking Mathematica for series expansions of certain hypergeometric functions. To give an example, consider the function $f(x) = {}_{5} F_{4}(3/2,3/2,3/2,2,2; ...
0
votes
1answer
47 views

Get dominant terms in asymptotic expansion

I'd like to find the dominant terms of an expression given the conditions $x,y >> 1$: $c_0 y+\frac{y^2 \left(4 c_0^2 y - 12 c_1 c_2^2+y^2\right)}{24 x^2 \left(\text{c1}^2+\text{c2}^2+y\right)}$ ...
1
vote
1answer
80 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
205 views

Plotting a Taylor series of Partial sum

Hi people, I want to plot the partial sum from n=0 to n =6 with the given function f about the point a=0 (just the maclaurin series). Unfortunately, I am unable to make sense of the issue M'tica is ...
0
votes
1answer
84 views

how can I convert a sum series to an equivalent product series

There are many identities in which on one side we have a summation series and on the other side we have a product series. For e.g. the very famous rogers-ramanujan identity ∑_(n=0)^∞▒q^(n^2 )/〖(q;q)〗n ...
17
votes
1answer
196 views

SeriesData sucks when it can. How do I keep SeriesData from sucking?

When I run Series[f[x]*Sin[x],{x,0,3}, Analytic->False] I get: f[x](x-x^3/3+O[x]^4) as expected. In ...
1
vote
0answers
52 views

How to erase the O[x] terms after using Series [closed]

I have a matrix in which I have used terms like: Series[Sin[(a qy)/2] , {qy, 0, 2}] How to get rid of the O[qz]^3 afterwards ...
4
votes
2answers
245 views

Determine the coefficient of expansion of the product of two sumations?

I would like to determine the coefficient of a desired term in the product of two summation where the powers of $x$ are not necessarily integers. For example $$ \sum_{i=1}^N x^{1/i}\sum_{j=1}^N ...
2
votes
1answer
69 views

Most simple example where series expanding a root object actually fails due to branchcuts

When computing the series expansion of a Root object Mathematica throws an error like: "Because of branch cuts, the series may represent a different root of [root expression] for some values of ...
2
votes
0answers
86 views

Peculiarities with Series and fractional exponents or bug?

The documentation of the Series[] function states that it can handle "certain expansions involving negative powers, fractional powers, and logarithms." What are the ...
2
votes
1answer
120 views

Solve an inequality involving a sum with a parameter

Good day. I am new to Mathematica and I am looking for advice. Is it possible to solve $$\sum_{k=0}^{n} \frac{p^k}{k!}>0$$ for $n$ where $p$ is a parameter? When I do ...
0
votes
1answer
67 views

What is the best way to generate this power series expansion?

f[m_,z_] := (1/m)*Sum[Exp[z*Exp[2*Pi*I/m]^k], {k,0,m-1}] g[t_] := 1/(2-f[5,t^(1/5)]) Series[g[t], {t,0,10}] When I tried to compute this on Wolfram Programming ...
0
votes
1answer
165 views

Taylor Series vs. Series Function [closed]

Could someone please explain why the these two functions give two different results ...
2
votes
2answers
155 views

SeriesCoefficients expansion contradicts FullSimplify

Executing the following SeriesCoefficient[Log[1/2 (1 + Sqrt[1 - x])], {x, 0, n}, Assumptions -> {n >= 1, n ∈ Integers}] I get: which clearly asserts ...
0
votes
4answers
77 views

Remove low orders from Series [closed]

It is easy to truncate Series upto some order, say $n$. My question is how do I remove low orders? Let us say my series is a power series in $x$. I want to remove the terms with negative powers ...
3
votes
1answer
47 views

Order of evaluation of Exp and Normal on result from Series

This may be more math related than Mathematica related, but I thought this might be of interest to the group. I'm trying to work with some Taylor Series approximations of functions that are ...
10
votes
1answer
287 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 ...
0
votes
1answer
98 views

second order nonlinear ode

I need to solve the following equation for $u_n(t)$: $u''=\frac{u'^2}{2u}+\frac{3u^3}{2}+4tu^2+2\left(t^2+\frac{n}{2}+(2p+1)\frac{1+3(-1)^n}{4}\right)u-\frac{n+(2p+1)(1-(-1)^n)}{4u}$ where $p$ is a ...
5
votes
2answers
234 views

Evaluating a Series expansion of PolyLog function

I am trying to evaluate an expansion of the following integral $$ \int_0^\infty \frac{p^4}{1+\exp\left({\frac{p^2}{2mT} - \frac{\mu}{T}}\right)}\, dp = A_0(m,\mu) + A_2(m,\mu)T^2 + \ldots $$ in terms ...
0
votes
0answers
14 views

Using InverseSeries on series with non symbol series parameter

I am attempting to use InverseSeries in the following way: InverseSeries[A[y]+A[y]^2+O[A[y]]^3, A[x]] However, upon execution it does not perform the operation, ...
7
votes
2answers
131 views

Generating terms of the Stirling series

The Stirling series starts as follows: $$n!=\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n} \left\{1+\frac{1}{12n}+\frac{1}{288n^{2}}-\frac{139}{51840 n^{3}}-\frac{571}{2888380 ...
-1
votes
0answers
58 views

Dirichlet series expansion

Is there a way to get Dirichlet series expansions in Mathematica? For example, I would want DirichletSeries[Zeta[s], {s, 4}] to return ...
4
votes
0answers
62 views

Quirky behavior of Series[]

This is a "what's going on?" question about MMa behavior, not so much a "how to fix?" This code calculates a Taylor series for a two-term Gaussian Mixture: ...
1
vote
1answer
62 views

Applying the common factor to each term of a series

I have a series like this: Sum[(n/z)^(1 - j + n)*Binomial[1 + n, j]*G[j], {j, 0, 1 + n}]/ ((1 + n)*(n/z)^n) where G[j] is, ...
0
votes
0answers
46 views

Constructing a function for expanding general $n$ products

I have the following quantum mechanically motivated product: $\langle0\vert(A_1b_1 + A_2b_2 + A_3b_3)(B_1b_1 + B_2b_2 + B_3b_3)(C_1b_1 + C_2b_2 + C_3b_3)$, where $b_i$ is an annihilation operator ...
0
votes
0answers
97 views

Series of square root of exponential

my problem concerns the Series command applied to Sqrt of an exponential, and it can be presented in a simplified version as ...
6
votes
1answer
321 views

Find exponential generating function from the first few terms

The function FindGeneratingFunction computes the ordinary generating function of a sequence, given a sufficient number of initial terms. I have two questions in ...
0
votes
1answer
42 views

Plotting partial sums of Taylor Function

I need to plot the partial function: $T_f(x)=\log 2+\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{2^kk}(x-2)^k$ for $n=2$,$n=4$ and $n=6$ on the same plot over the interval $[-\frac{1}{2},4]$ and also the ...
1
vote
1answer
180 views

Evaluate $\pi$ using While

I am given the following exercise: Evaluate $\pi$ using the formula $\pi = 4 - \sum_{n=1}^{\infty} (-1)^n \frac{4}{2n +1}$ with precision $\epsilon = 0.0001$. First I defined the function I need ...
0
votes
1answer
53 views

Series of integrate in exponential

Why Mathematica cannot perform this series: ...
3
votes
0answers
119 views

Discrepancy in the series expansion of $\log(z/(z-1))$ in Mathematica 5

When I calculate the series expansion of $\log\frac{z}{z-1}$ around $0$ in Mathematica 5.2, I obtain: ...
0
votes
0answers
37 views

Series Expansion of Differential Operators [duplicate]

We know about the possibility of series expansion of a function in Mathematica. Same as below, where a function f[a x] is expanded around zero up to 5th order. ...
4
votes
2answers
258 views

Taylor series representation as an infinite sum

I want to see the Taylor series representation for arbitrary functions, e.g. $\sin$. With the Series[] command, I can only see the first $n$ terms. Is there the ...
0
votes
1answer
71 views

Strange long time evaluation of Series for fractional function

my problem concerns the Series command applied to a product of a rational function times a square root. This can be exemplified in the following way ...
2
votes
0answers
126 views

Puiseux series for algebraic curves

Has anyone implemented a function in Mathematica that computes Puiseux expansions of algebraic curves? Using something like ...
-2
votes
3answers
71 views

Two unrelated questions: 1. Output of Series[] for function, 2. Output conditional answers as if not conditional [closed]

As mentioned in the title, I have two questions. 1. I have code that looks similar to: ...
13
votes
3answers
626 views

How can I obtain an asymptotic integral expansion at infinity?

I want to find the asymptotic expansion at $x \to \infty$ of the following function: $$ I(x) \equiv \int_0^{\pi/2} e^{-xt^3 \cos(t)} dt.$$ To do this I defined ...
1
vote
1answer
78 views

Subtracting Series

When I input the following $$\sum_{n=0}^{m+1}x[n]-\sum_{n=0}^{m}x[n]$$ which in InputForm is: ...
0
votes
1answer
56 views

How to turn arbitrary function to polinomial series in Mathematica?

Can I turn any multivariate function into polinomial series in Mathematica? Suppose I have a function Fwd[x_, α_] := x (1/Sin[Pi x/2])^α and wish to express it ...
6
votes
2answers
251 views

How to reverse irreversible function in Mathematica?

How to reverse formula $y(x)=x (\frac{1}{sin \frac{\pi x}{2}})^\alpha$ i.e. express it as $x = x(y)$ in Mathematica? I did this way ...
0
votes
0answers
47 views

FourierSinCoefficients extremely slow

I'm just playing with Fourier decomposition of periodic functions. We had a similar example with standing waves and cosines in class, so I tried to expand a triangle wave in sines. This is what I came ...