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

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54 views

Asymptotic solution

I have an ODE V''[z] + ( (I z0^2 w)/(3 (z - z0)^2) + (2 - (4 I z0 w)/9)/(z - z0)) V'[z] + (-(12/41) I z0 w + (23 z0^2 w^2)/369)/(z - z0)^2 V[z] == 0 ...
3
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1answer
85 views

Speedier Algorithm or Set of Steps to Increase Speed of Coefficient Generation

Good morning. I am working with higher powers of the Generalized Euler Number generating function $$\left[\frac{n}{\sum_{j=0}^{n-1}\exp{\left(w_n^jx\right)}}\right]^\alpha$$ where $w_n=\exp{\left(...
1
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2answers
85 views

Engel expansion

How do I do an Engel expansion in Mathematica? And find the unique non-decreasing sequence of positive integers $ \{ a_1 ,a_2,a_3,\dots \}$ , such $$\frac{e}{\pi}=\frac{1}{a_1}+\frac{1}{a_1 a_2}+\...
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0answers
28 views

How do I do Cauchy product of infinite series?

I have two series and want their Cauchy product series as a result. For instance, I want to multiply Sum[1,{x,0,Infinity}] to Sum[2,{x,0,Infinity}] The directConvolve method does not work, ...
5
votes
5answers
188 views

How to find the derivative value at $(\pi,0)$ for this implicit function $n$ times?

I am trying to take the implicit derivative at $\sin(x+y)+\sin(x)=y$ and substitute $x=\pi$ and $y=0$ at least 6-7 times since I need to find the Taylor series for this function. Since I barely ...
3
votes
2answers
61 views

Getting rid of sub-exponential terms in an asymptotic expansion for a modified Bessel function

I am trying to get Mathematica to produce suitable asymptotic expansions for some modified Bessel functions at large argument (more specifically, the expansion in the DLMF's eq. (10.40.1)), and I'm ...
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0answers
64 views

How can I verify this limit? [closed]

Hello I have to verify this limit $$ \lim_{N \to \infty} \frac{2 \pi}{N+1}\sum_{n=0}^{N}\sin^2(q_n)\frac{1}{E_n}\left [ \delta(\omega -E_n)+\delta(\omega+E_n) \right ] = \mathrm{sgn}(\omega) \Theta(1-...
6
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0answers
100 views

Getting around Series[Sinc] bug

Bug introduced in 8.0 or earlier and fixed in 10.4 For some reason, Series expansion of Sinc around ...
1
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1answer
70 views

How can I make Mathematica list the terms in this series?

I am using perturbation theory to solve a problem of the following form: $$ R(h,\theta)f(\theta) = h g(\theta) = 0 $$ where $h$ is small, and I assume $$ \theta = \sum_{i=0}^\infty \theta_i h^i $$ ...
3
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1answer
124 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 ...
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0answers
65 views

Interpretation of a function

I know this is not great, as far as a question, but I came across this function, ...
7
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6answers
3k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
3
votes
1answer
55 views

Differentiation and series expansion of dot product - inconsistent results

If I differentiate a dot product, I get the result I expect D[a.b[x], x] (* a.b'[x] *) However, a series expansion of the same expression gives a very different ...
1
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2answers
75 views

Series of a hypergeometric function

Let $n>2$ be odd, and let $x\in [0,1]$. I would like to calculate the Taylor expansion of $$ x^{2-n} \, _2F_1\left(-\frac{n}{2}-1,-n;2-\frac{n}{2};x^2\right) $$ at $x=1$ leaving $n$ non specified....
5
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1answer
263 views

Finding an analytic solution of a cubic equation

I have been trying to solve a cubic equation $$y=zy^3 + z$$ in $y$ – that is, my desired result is a function $y(z)$ satisfying the equation. Now, there are three solutions of this equation and ...
1
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2answers
159 views

How do I get Series[] as a functional, rather than as an expression (i.e. to avoid the dummy variable)?

How can I write an equivalent to Series that doesn't require a dummy variable? Note that the series should be constructed before the evaluation point is supplied, ...
1
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1answer
73 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; 1,...
5
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2answers
256 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 ...
1
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1answer
81 views

Weird SeriesData behavior?

I was trying to calculate the determinant of a matrix whose elements are truncated series. As a result I obtained an expression like this: ...
7
votes
2answers
141 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 $k$....
4
votes
1answer
112 views

Series expansion of $(x;x)_\infty$ at $x=1^-$?

The so called Euler function is implemented in Mathematica as QPochhammer[x, x] I have been trying to obtain its leading behavior for $x\to 1$ from below. ...
1
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1answer
64 views

Automation of Perturbation Solution

I'm trying to solve an equation of the form $$ R(\theta)f(\theta)+hg(\theta) = 0 $$ for small $h$, where $R$, $f$, and $g$ are functions. I've assumed a power series expansion for $\theta$ in terms ...
2
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2answers
165 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 ...
2
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3answers
922 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 ...
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0answers
42 views
2
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3answers
1k views

Calculating Taylor polynomial of an implicit function given by an equation

I'd like to write a procedure that will take an equation: F(x,y,z) = 0 chosen variable: x a point: ...
3
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3answers
175 views

OEIS A144311 Generating function

I'm looking for a way to use calculate OEIS A144311 efficiently in Mathematica. First, let's define the series. In one sense or another, this series considers the number between "relative" twin ...
4
votes
1answer
208 views

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

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

Asymptotic forms of Bessel function

I want to replace Bessel functions by asymptotic forms, so the question is: can I find the best ones with help of Mathematica? And if it's possible, how can I do it? Update How can I get with ...
3
votes
0answers
43 views

Series expansion in Infinity issue with Zeta(s) function

With this code: Series[Zeta[s], {s, Infinity, #}] & /@ Range[10] // MatrixForm Series expansion for the Zeta(s) ...
10
votes
3answers
472 views

Chebyshev Approximation

Is there functionality in Mathematica to expand a function into a series with Chebyshev polynomials? The Series function only approximates with Taylor series.
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1answer
38 views

Maclaurin Series Help [closed]

My problem is to numerically approximate the series (1 - Cos[x])/x over the interval [0,1]. I typed it into Mathematica as so: ...
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3answers
55 views

Can highlight an expression in mathematica in an expansion?

I ask mathematica to make an expansion of some expression. Is there a way to ask mathematica to highlight all terms that, say, goes by x^3 since it is really hard to find them? E.g. ...
1
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0answers
242 views

A power series expansion [closed]

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 $n \in \mathbb{Z}$ ...
1
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0answers
77 views

Ramanujan's asymptotic formula for $p(n)$ [closed]

The following expression is the exact formula of partition function. Ramanujan's asymptotic formula for $p(n)$ is following $$p(n)\sim\frac{1}{4n \sqrt{3}}e^{\pi \sqrt{2n/3}}$$ Can I use ...
4
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0answers
86 views

Bug in Series[Pochhammer[1+n, n], {n,Infinity,1}]?

The function Pochhammer[1 + n, n] tends to infinity. We have ...
1
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1answer
79 views

Find the function $f(x)$ by using its fourier expansion

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem by using Mathematica How to find the function $f(x)$, if I know its ...
2
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0answers
44 views

Unexpected imaginary term in the asymptotic expansion of DawsonF

FullSimplify[Series[2 DawsonF[x], {x, ∞, 8}]] (* -I Exp[-x^2] √π + (1/x + 1/(2 x^3) + 3/(4 x^5) + 15/(8 x^7) + O[1/x]^9) *) What is the reason the term ...
1
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1answer
30 views

`Series` acts on variables outside its argument?

I am trying to work with Taylor series. What I am trying to do is to express $$ f(h) - f'(0) = f(h) - f'(h - h) $$ such that only derivatives of $f$ with the argument $h$ appear, for example $$ f(h) - ...
4
votes
1answer
153 views

Why does merely writing SeriesData[] with zeros give an error, even if it is never called?

Bug introduced in 8.0 or earlier and persisting through 10.4.1 Why does the code Function[t, SeriesData[t, 0, {0}, 0, 1, 1]] give me the following error? <...
3
votes
1answer
27 views

Asymptotics of Bessel function for real arguments

I am trying to calculate the following asymptotic behaviour: Normal@Series[BesselK[1, r \[CapitalLambda]] / BesselK[1, \[CapitalLambda]], {r, 0, 1}] but for $\...
2
votes
1answer
65 views

Series expansions and algebraic branch points

I have the expression $$(1-x^4)^{-1/4}$$ where $0<x<1$, which gives me a real number. When I try to expand it around $x=1$, Mathematica gives me complex coefficients, which makes me think that ...
4
votes
1answer
126 views

Successive Series Expansion Bug (?)

I've found what appears to be a bug in MMA related to taking successive series expansions. I'm providing this minimal example and post as other posts didn't appear to address the issue I found. In ...
0
votes
2answers
104 views

Solving recursion relation from power series

I am interested in solving differential equations in the form of power series. Let's say we have following equation: $$f^{\prime \prime} (\rho) + \left( \frac{2 e^{-k \rho}}{\rho} - \varepsilon \...
3
votes
0answers
72 views

A series in powers of $(a-z)$ instead of $(z-a)$

Sometimes it is more convenient to find a series expansion (e.g., Taylor, Laurent, Puiseux, ...) in powers of $(a-z)$ than in powers of $(z-a)$. For instance, the command ...
0
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0answers
56 views

break down the function

I am using Series function, but I found that it cost too much time. Since the Series function is Taylor expansion and I only need first order, I want to break down the code, so I can make it faster <...
0
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0answers
60 views

Sum of Infinite convergent series

I have a series as Sum[(x^1 - (x - 1)^1) (b^Log[(1 + (x))]), {x, 1, Infinity}] which is convergent when b=0.3 and ...
1
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2answers
67 views

How to expand an infinite product?

How can I use Mathematica to expand such a product (only need a finite number of terms): $$\prod^{\infty}_{n=1}\frac{({1-yq^{n+1}})({1-y^{-1}q^n})}{(1-q^n)^2}$$
1
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1answer
117 views

Problem with solving a system of differential equations [closed]

I am trying to reproduce a result that is part of a derivation of the flow due to a rotating disk. I have as given this system of differential equations. (In these equations primes indicate ...