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

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0
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1answer
59 views

Replacement of terms/Pattern matching involving products of derivatives of a function

In delving into Ramanujan summation, I'm trying to get a hold of the relations of the form $$\sum_{n=0}^\infty f(n)=\dfrac{h\frac{d}{dx}}{{\mathrm{e}^{h\frac{d}{dx}}}-1}\int_{0}^\infty ...
0
votes
1answer
39 views

How to obtain Laurent series coefficients of an (almost) arbitrary function?

I have observed that Series in Mathematica assumes that the given function is smooth in the point around which one wants to perform series expansion. For instance: ...
2
votes
4answers
83 views

Extract a part of Series

If I have the output of Series, in terms of powers of my variable $x$, what is the quickest way to extract a part of the series, say for example the terms from $x^2$ to $x^5$, excluding those with ...
1
vote
1answer
32 views

Truncate a sum in a 'smart' way

Consider a series like this: $$\sum_{n=1}^{\infty} (c_n r^{2-n/2}+d_n r^{-n/2})$$ Sum[c[n]r^(2-n/2)+d[n]r^(-n/2),{n,Infinity}] I want, now, to keep only the ...
0
votes
1answer
22 views

Extract a term of `Series` output

Say I have the output of series, with all the coefficients of the different powers of my variable $x$. What is the quickest way of extracting the coefficient of the $n$-th power of $x$?
0
votes
1answer
43 views
2
votes
2answers
69 views

Series expression of a Root object

I was wondering if it is possible to get Mathematica to return a series approximation of a Root object. Example: I want a series representation of x in terms of ...
1
vote
2answers
103 views

Linearization of differential equations

I was wondering if one could define an operator such that, when I give a certain number of (differential) equations as an output, and an "equilibrium" value for each of the variables, it returns the ...
1
vote
1answer
89 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 ...
3
votes
1answer
131 views

Series approximation to integral

I would like to approximate the integral $$ \int_0^\infty dy\,\frac{1}{\sqrt{2\pi y\sigma^2}}\exp\left(-\frac{(x-y)^2}{2y\sigma^2}\right)f(y), $$ as a series expansion in the limit $\sigma\rightarrow ...
5
votes
5answers
117 views

Isolating cross terms

I have a set of expressions that contains terms like $\frac{1}{(x + a + b)(c+d)}$. I would like to simplify the denominator so that products of $a,b,c,d$ are dropped, but $x c$ and $x d$ are kept. ...
0
votes
1answer
60 views

Can't get solution to a system of recurrence equeations

I would be grateful if someone could help me with this problem. I define two series recursively, thus: ...
0
votes
0answers
49 views

Linearization of differential equation - need tips for the use of Series function

I would like to linearize a differential equation around a equilibrium position. The description of the steps that I have carried out are : Equation ...
0
votes
0answers
99 views

Alternative to Series

Here is a sample of my code: ...
7
votes
1answer
217 views

Error in infinite sum

The binary weight of the non negative integer k is defined by w[k_] := Total[IntegerDigits[k, 2]] The first values are (cf. http://oeis.org/ A000120) ...
1
vote
2answers
36 views

Imploying a sequence in a series

I would like to have the following sequence, $(a)_n$, in a series where $a_0 = 1$ and $a_n = a(a+1)\cdots (a+ n -1)$. The series I have is $$ 2\sum_{n=0}^{\infty}\frac{(1/2)_n}{n!(2n+1)^3} $$ I know ...
0
votes
1answer
29 views

Derivative inside a Series [closed]

I have a function defined by a sum, which I would like to differentiate. For example: ...
1
vote
1answer
46 views

Don't understand why my Taylor expansion results in a message and an unexpected result

Trying to do something simple: Taylor expand a generic function of t around a point t and substitute ...
5
votes
2answers
81 views

Replace expression in series expansion [duplicate]

A test case: I'm trying to replace an expression inside a series expansion: Series[f[x],{x,x0,4}] ./ (x-x0)->h but it still returns ...
13
votes
4answers
4k 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 ...
0
votes
1answer
79 views
10
votes
2answers
433 views

InverseSeries of multiple variables and multiple equations

CONTEXT Let us consider a bit of the Universe in which we draw spheres (see a high resolution image here). Astronomers have shown that the density within these spheres could be predicted quite ...
0
votes
1answer
43 views

Series expansion of large expression

I have the following equation: eq=E0[x,y,z]+E1[x,y,z]*Cos[phi]+E2[x,y,z]*Cos[phi]^2+E3[x,y,z]*Sin[phi] Now E0,E1,E2 and ...
0
votes
0answers
31 views

SumConvergence with product $\sum_1^\infty{\frac{1\times3\times5\times…\times(2n-1)}{n!}}$

SumConvergence[( Product[(2 n - 1), {n, 1, infinity}])/n!, n] $$\sum_1^\infty{\frac{1\times3\times5\times...\times(2n-1)}{n!}}$$ However this returns true but it ...
0
votes
2answers
55 views

Failed to use N[%] for a infinite series

I am trying to evaluate a infinite series: $$ 2 \sum _{l=1}^{\infty } \Re\left(n^{\frac{2 i \pi l}{\log (q)}} \Gamma \left(\alpha -\frac{2 i l \pi }{\log (q)}\right)\right) $$ With the code: ...
1
vote
2answers
77 views

Series with specific notation

I'm trying to get mathematicas series function to output a result that look like this: instead of calculating the actual "values", like doing so here: ...
2
votes
3answers
234 views

Sum the coefficient of a series

I am computing the Series expansion (Lauren series) of an integral and I want to sum up the coefficients of the series. ...
2
votes
1answer
136 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 ...
1
vote
1answer
37 views

Series power with unusual denominator

I have a series whose terms are: (x^n)/(n+a) where n is a positive integer, x a real number that is greater than 0 and smaller than 1, and a is a real number smaller than 1. It is easy to proof that ...
3
votes
1answer
37 views

Defining the value of variable after expansion

I have an initial equation defined as: x = Subscript[a, 0] + (1 - r^2)/Sqrt[1 + r^2 - 2*r*Cos[\[Theta]]]; I want to taylor expand this after subbing in: ...
3
votes
1answer
99 views

List interpolation

Hi mathematica people! So i am looking for the best way to interpolate a function given a list of its values. I have an iterative algorithm which needs high precision otherwise the numerical noise is ...
0
votes
1answer
74 views

How to obtain Padé approximant of $\log(x)$? [closed]

Having read this answer on Math.SE, I wanted to try seeing how Padé approximations converge to $\log(x)$. But my first attempt after reading the documentation failed: ...
4
votes
3answers
198 views

Polynomial with alternating sign coefficients from the odd degree terms of a Truncated Power Series

I'm trying to make a function that will take any truncated power series I give it, strip away all the even degree terms, then change the sign of the coefficients of every other term. So that we have a ...
0
votes
1answer
56 views

Single formula for this sequence [closed]

Consider the following sequence. 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8 We can express this sequence by (an) where an=n/2 when n is even and an=0 when n is odd. Find a single formula for an ...
2
votes
1answer
309 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: ...
2
votes
1answer
91 views

Evaluate function defined by DifferentialRoot

I have the following sequence of rationals that I want to find the generating function of: ...
1
vote
1answer
79 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 ...
4
votes
1answer
151 views

How do I find a series solution for $$e^{-\frac{1}{2}f'(x)} cosh( f(x) ) = ax + b$$

I am trying to approximate a function $f(x)$ satisfying a relation between $f(x)$ and its first derivative. How do I find a series solution for $$e^{-\frac{1}{2}f'(x)} cosh( f(x) ) = ax + ...
3
votes
1answer
113 views

Problem with series expansion and integrate

I have the following very simple code. A power series in a-parameter, function f[x], is integrated with respect to x, and the ...
14
votes
1answer
246 views

Why do big-O terms disappear in definite integrals since Mathematica 9?

In Mathematica 8, when I computed the following input: Integrate[Series[Cos[x], {x, 0, 2}], {x, 0, a}] Mathematica returned an expression that had a O[a^4] in ...
4
votes
0answers
53 views

FindSequenceFunction for sum of hypergeometric terms

Mathematica's built-in function FindSequenceFunction is quite good at recognizing hypergeometric terms, i.e. terms $c_k$ for which $c_{k+1}/c_k$ is a rational ...
2
votes
3answers
206 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 ...
2
votes
1answer
60 views

Truncating power series

I have the following typed up so as to truncate higher powers of $\tau$, ...
1
vote
1answer
81 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
66 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
97 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 ...
0
votes
1answer
63 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
780 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
1
vote
1answer
98 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 ...