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

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0
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0answers
26 views

Proof bu induction? [on hold]

I've been stumped on this for a while, and even had a EE PhD take a look with no joy. show that for any n >= 2, the following formula is valid: 1^2 - 2^2 + 3^2 - ... + (-1)^(n-1)n^2 = ...
16
votes
0answers
139 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
43 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 ...
6
votes
6answers
2k 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 ...
4
votes
2answers
172 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
52 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 ...
3
votes
1answer
2k views

Laurent series expansion

Can someone share how to find a Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at zero on the annular disk $1<|z|<2$?
2
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0answers
43 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 ...
4
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1answer
197 views

Problems with series of generalized hypergeometric functions

I have been trying to do the following series: Series[HypergeometricPFQ[{1, 4, 4}, {4 - Sqrt[3], 4 + Sqrt[3]}, z],{z,1,0}] Mathematica says that the result is ...
6
votes
1answer
212 views

How to compute the residue of $e^{z-\frac{1}{z}}$ at z=0?

I've tried the following but it didn't work: Residue[Exp[z - 1/z], {z, 0}] not even this: Residue[Exp[1/z], {z, 0}] ...
14
votes
1answer
1k 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 ...
0
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0answers
191 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 ...
9
votes
1answer
188 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 ...
6
votes
1answer
230 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 ...
6
votes
1answer
89 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 ...
2
votes
1answer
99 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
37 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
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1answer
143 views

Taylor Series vs. Series Function [closed]

Could someone please explain why the these two functions give two different results ...
2
votes
2answers
147 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 ...
10
votes
1answer
188 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
4answers
47 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 ...
2
votes
2answers
132 views

The proper way to write the input for a certain series

Mathematica tells the series below doesn't converge. I think it converges. What would the proper way to write things be as an input? ...
3
votes
1answer
43 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 ...
0
votes
1answer
91 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 ...
4
votes
2answers
110 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 ...
5
votes
2answers
195 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 + ...
0
votes
0answers
10 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, ...
3
votes
1answer
56 views

Generating terms of the Stirling series

The Stirling series starts as follows: $$ n!=\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n} \Bigl\{1+\frac{1}{12n}+\frac{1}{288n^{2}}-\frac{139}{51840 n^{3}}-\frac{571}{2888380 n^{4}}+O(n^{-5})\Bigr\}. $$ ...
0
votes
1answer
275 views
0
votes
0answers
43 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
53 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: ...
2
votes
1answer
427 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: ...
1
vote
1answer
57 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
1answer
56 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 ...
1
vote
1answer
108 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 ...
0
votes
0answers
42 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 ...
12
votes
1answer
349 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}]
3
votes
2answers
262 views

Bessel derivatives

I tried to expand BesselJ[k,x] function into a Taylor series with Series command. Here both ...
0
votes
0answers
56 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 ...
1
vote
1answer
145 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
37 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 ...
12
votes
3answers
395 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 ...
0
votes
1answer
52 views

Series of integrate in exponential

Why Mathematica cannot perform this series: ...
3
votes
0answers
93 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
143 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 ...
18
votes
4answers
476 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 ...
-2
votes
3answers
65 views
2
votes
0answers
57 views

Puiseux series for algebraic curves

Has anyone implemented a function in Mathematica that computes Puiseux expansions of algebraic curves? Using something like ...
0
votes
0answers
33 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: Sum[x[n], {n, 1, 1 + m}] - Sum[x[n], {n, 1, m}] it returns ...