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

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5
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4answers
436 views

Exponential of a Differential Operator

In Mathematica, is it possible to exponentiate a differential operator such that the operator will act on a function, $f(x,p)$? Specifically, I wondering if I can get Mathematica to do this: ...
0
votes
1answer
146 views

How to power-series expand determinants?

Say $g$ is a matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and ...
0
votes
3answers
94 views

Create polynomials from Series

This question actually doesn't have quite a lot to do with the Series function, but I don't know how to describe my problem. So here's the thing. I'm trying to ...
12
votes
1answer
347 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}]
2
votes
1answer
139 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
73 views

Series and that old ivar error

I am trying to compare plots of the function (1+x)^{1/3} and its power series. The code for the power series generates an error message referring to an invalid variable: 'General::ivar: -1.09993 is ...
2
votes
1answer
113 views

Remove singularity at zero from $-((i^{-n} (-1 + i^n)^2)/(n^2 π^2))$

The expression h[n_] = -((I^-n (-1 + I^n)^2)/(n^2 π^2)) is real for all integers n. Although indeterminate at ...
1
vote
1answer
147 views

Working with means and variances

My goal is to work out on the following equation: $$Mean(P)(1-Mean (P))\left (h + (1 - 2 h) (1 - \frac {Var (P)} {Mean (P) (1 - Mean (P))}) (1 - Mean (P)) + \frac {Var (P)} {Mean (P) (1 - Mean ...
2
votes
1answer
165 views

Taylor expansion of function of vectors—simplifying the output

I've got a real-valued function of several vectors $f(u,v,w)$ formed by taking scalar products of linear combinations of the vectors, I want to Taylor expand around small $v$ by writing $$f(u,\delta ...
1
vote
1answer
79 views

Compose two special power series expansions

I have two functions $A(x), B(x)$, given in a special power series form: $A(x)=1-x^{2}\left(\frac{a}{10}-\sum_{k=1}^{9}b_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$ ...
3
votes
2answers
259 views

Bessel derivatives

I tried to expand BesselJ[k,x] function into a Taylor series with Series command. Here both ...
0
votes
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 ...
1
vote
1answer
108 views

About doing an sum

I want to explain this summation to Mathematica, For $a >0$ define $n_u, n_d \in \mathbb{Z}$ such that if $\{ a \} \leq 0.5$ then $n_u = [a]-1$ and $n_d = - [a]$ and if $\{ a \} > 0.5$ then ...
6
votes
1answer
123 views

Is it possible to circumvent a bug inside SeriesCoefficient?

As far as I can tell, there seems to be a bug in SeriesCoefficient: ...
2
votes
1answer
423 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
127 views

How to expand function $\cos(y+i\log{x})$ in powers of $x$?

I have the following, probably very simple question. How can I get $\it{Mathematica}$ to power expand function $\cos(y+i \log(x))$ in powers of $x$? This function obeys a well defined Laurent ...
0
votes
1answer
267 views

How to simplify power series in terms of functions

For example, after some computations mathematica outputs $$ \sum_{k=0}^{\infty} \frac{k g_k z^{k+4}}{(1-z)^3} $$ and we assume that $$G(z)= \sum_{k=0}^{\infty} g_k z^k $$ so the question is: how ...
0
votes
1answer
90 views

How could I expand this equation

How could I expand this equation (0.837 + x^0.5 + x)^4.870 Neither Expand[(0.837 + x^0.5 + x)^4.870] nor ...
0
votes
1answer
81 views

nested series expansion

I have to invert the following matrix in which the functions U[t,x,y,r] and K[t,x,y,r] and all their derivatives are "small"; ...
0
votes
1answer
199 views

Multivariate series expansions to different powers

I have a function that (I believe) correctly takes the multivariate Taylor series expansion about the origin for some expression (first argument), in some variables (second argument, list), to ...
0
votes
2answers
160 views

Expand expression into quotient of negative power series

I have a function T(z) and want to expand it to the following form: At the end I want to have a list of equations, which can calculate a0, a1, a2, b0, b1 and b2 with different values of h1, h2, ...
1
vote
2answers
159 views

Making mathematica do regulated integrals

Consider the integration, $\int _0 ^\infty dx\ x \tanh( \pi x) \sqrt {x^2 + a^2 } $ where $a$ is a real number. This integral is divergent. We note that an asymptotic expansion of the integrand ...
3
votes
1answer
163 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 to ...
13
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
votes
1answer
103 views

Evaluate integral of a series

I want to evaluate this integral, but it won't work. Does anyone knows why? ...
5
votes
1answer
590 views

Asymptotic expansion

I wanted to expand a function of $x$ about $x=\infty$ and see its coefficients as a function of the parameters $m,n,q,y$ and I wrote this - but it didn't work! It gave me back a very complicated ...
5
votes
2answers
181 views

How to get the series expansion of $e^{x^a}$ at $x=0$?

I want to have a series expansion of $e^{x^a}$ or $e^{c_1x^a+c_2x^b}$ at $x=0$, but Series cannot give any useful result even if the assumption $a>0$ is ...
5
votes
3answers
355 views

Series coefficients for an infinite sum?

I'm working through Carl Bender's Mathematical Physics lectures on YouTube (which are great fun), and I'd like Mathematica's help solving terms in the perturbation series. It would be convenient if ...
2
votes
2answers
69 views

Finding series expansion in parametric setting

Suppose $y=f(x;p_1,p_2,...)$ and $t=g(x;p_1,p_2,...)$ are given where $p_1,p_2,...$ are some parameters. We want to find a series expansion of $y$ in terms of $t$. Is there a direct command or ...
1
vote
1answer
411 views

Series expansion using an exponential basis

I am currently trying to Laplace invert an expression with the following pattern $$ \frac{s \alpha \text{Cosh}[s(L-x)]+\beta \text{Sinh}[s(L-x)]}{s(\gamma \text{Cosh}[sL]+s \delta \text{Sinh}[sL])} ...
4
votes
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 ...
3
votes
1answer
595 views

How to get rid of all complex numbers and functions?

For almost esthetical reasons, I always have been fascinated by infinite sums of series and I always wonder if theyhave a closed form. A couple of days ago, I found on MSE a question related to the ...
1
vote
1answer
77 views

Reducing a multi-variate polynomial to have terms upto certain degree

I have a polynomial in x and y. I want to keep all the terms with (combined) degree 4. Any pointers will be appreciated. Simple tricks for doing the same with univariate polynomial don't seem to work. ...
6
votes
1answer
211 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}] ...
1
vote
1answer
73 views

How can we tell Series to treat the expansion parameter as an integer?

It is simple for Mathematica to find an asymptotic expansion for $\frac{1}{-1+p}$ as $p \rightarrow \infty$. However, if we want to restrict $p$ to be an integer and also include some terms that ...
6
votes
3answers
538 views

How to make all numbers equal to one in a Series?

I have many outputs of one-dimensional Series expansions for which I am only interested in the general tending with the variable. For example, I would like to be able to transform something like ...
0
votes
2answers
215 views

Asymptotic series [duplicate]

I need to solve the following problem. I have the following function: Erf[Sqrt[a x^2 + b y^2]]/Sqrt[a x^2 + b y^2], where x and y are variables and a and b ...
-1
votes
1answer
73 views

equidistant solutions in Sequence Sums That Are Squares

In my demonstration "Sequence Sums That Are Squares" I demonstrate two "lines" of solutions (of infinite length) and ask whether someone can find one more line. I am looking forward to your solution. ...
1
vote
2answers
360 views

Expansion with binomial coefficients

How can I get binomial coefficients in expansion of $(n x+i) (1+i x)^n+(n x-i) (1-i x)^n$, where $i=\sqrt{-1}$ and $n$ is an integer. I have no idea how to coax Mathematica to do something remotely ...
0
votes
2answers
164 views
9
votes
0answers
358 views

How does Mathematica find a series expansion of expressions containing logarithms when there is a singularity at the expansion point?

I am looking for a good approximation to a function containing logarithms, especially at values close to zero. When I used Mathematica's Series command I found ...
-2
votes
1answer
86 views

Using Product for different variables

when using product in mathematica the initial and maximum value of the product will be the variable itself that will be maupulated. what I need is to make these initial and maximum of the product as ...
6
votes
1answer
91 views

Is it possible to find a function from first few terms in the expansion

Is it possible to find a function if first few terms of the expansion is known. For example if I have this series $f(x)=\frac{k^3 x^2}{6}-\frac{k^5 x^4}{120}+\frac{k^7 x^6}{5040}-\frac{k^9 ...
4
votes
3answers
121 views

Get interval of series

It seems to be a stupid question but I wonder how to get an interval of a series expansion. The current series command Series[f, {x, x0, n}] only give series ...
4
votes
1answer
248 views

Computing a series in terms of exponential function

Is there any way to compute the following series in terms of exponential function ? $$\sum_{k=0}^\infty Y_1(k)\;x^k$$ where $$Y_1(k) = \frac{(k - 1)!}{k!}Y_3(k - 1)$$ $$Y_2(k) = \frac{(k - ...
2
votes
1answer
134 views

Generating a power series expansion of the function with parameter

I'm trying to generate a power series expansion of the following function: ...
3
votes
2answers
147 views

Change all values of a series to positive values

I would like to change all values of a series to positive values. I refer to my previous question, answered in the most part, very elegantly by RunnyKine. However, there remains a slight glitch that ...
4
votes
1answer
179 views

List manipulation - various

Update OK, Please forgive the messiness of this, but I am working with something like: ...
2
votes
0answers
136 views

Series expansion: Taylor series takes huge amount of time

I'm working on a notebook, trying to expand the root of a cubic polynomial in Taylor series. When I type: ...
5
votes
1answer
117 views

Apparently contradictory output from Series and D

I defined a function as: g[b_] := Integrate[f[n]Exp[-b DA (n.u)^2], n] Obviously D[g[b], b] /. b -> 0 gives ...