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

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-2
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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
685 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
81 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
58 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
262 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
49 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 ...
10
votes
1answer
295 views

Contour Integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
0
votes
1answer
124 views

Series with a specified number of terms

I'm doing calculations with Series where I don't know the power of the leading order term. I would like to keep a specified number of terms, but since I don't know the leading order this is proving ...
20
votes
4answers
626 views

Expansion of a hypergeometric function takes ages with Mathematica 9 and 10 (regression?)

Mathematica 8 (Linux version) can evaluate AbsoluteTiming[Series[Hypergeometric2F1[1, 1 - eps/2, 3 - eps, 1/2], {eps, 0, 0}]] in no time. On one of the ...
1
vote
1answer
115 views
0
votes
1answer
188 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
102 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 ...
0
votes
1answer
42 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 $x$?
1
vote
1answer
100 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
81 views

Series identity with binomial coefficients [duplicate]

I have this apparently simple equation: ...
2
votes
2answers
153 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 <...
2
votes
2answers
599 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 ...
0
votes
1answer
82 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: ...
5
votes
5answers
182 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. ...
4
votes
1answer
275 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 ...
8
votes
1answer
301 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
39 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
79 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
100 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
105 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 ...
0
votes
0answers
131 views

Alternative to Series

Here is a sample of my code: ...
0
votes
1answer
122 views

Series expand root of quartic polynomial containing many real parameters and FullSimplify return `Indeterminate`

Using Mathematica I can get the Eigenvalues of the following matrix (using Quartics->True): ...
0
votes
1answer
69 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
33 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
68 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
81 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
423 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. ...
1
vote
1answer
49 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
42 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
211 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 ...
4
votes
3answers
287 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 ...
2
votes
1answer
207 views

Evaluate function defined by DifferentialRoot

I have the following sequence of rationals that I want to find the generating function of: ...
6
votes
2answers
243 views

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

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)} \mathrm{cosh}( f(x) ) = ax + e^{-...
1
vote
1answer
141 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 ...
14
votes
1answer
277 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 ...
7
votes
2answers
136 views

`FindSequenceFunction` for sum of hypergeometric terms?

The 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 function of $k$....
3
votes
1answer
202 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 ...
2
votes
3answers
910 views

Partial fraction decomposition of $1/(e^x-1)$

This link has discussion on finding partial fraction decomposition of $1/(e^x-1)$, so I experimented with Mathematica to see if M can do it, but looks like not. Similar is the case with one more CAS I ...
2
votes
1answer
107 views

Truncating power series

I have the following typed up so as to truncate higher powers of $\tau$, ...
2
votes
3answers
386 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 $k$...
1
vote
1answer
243 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
128 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
119 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 x,\...
1
vote
1answer
118 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 ...
2
votes
2answers
162 views

How to get `Series` to recognize user-defined function has pole

Edit for clarity: How does Mathematica's function Series know that Gamma[x] has a pole at ...