Questions tagged [series-expansion]

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

Filter by
Sorted by
Tagged with
5
votes
2answers
282 views

Why can’t mathematica find this residue?

I am trying to find the residue of the function $$f(z)=(z+1)^2e^{3/z^2}$$ at $z=0$. I tried the following in Mathematica Residue[(z+1)^2*Exp[3/z^2],{z,0}] which ...
0
votes
0answers
58 views

How can I select the true expansion?

I have a series expansion which gives two results. Series[-(1/ Sqrt[-1 + x + x z1^2 \[Chi]^2]) , {x, 0, 1}] /. y -> x I want only the series which is true. ...
3
votes
1answer
292 views

Series expansion really slow

UPDATE Seems the problem is associated with the division. I've tried taking the Normal[Series[]] of each numerator and denominator separately, and then again ...
2
votes
2answers
72 views

Grouping terms in Taylor expansion

I have this code: ...
4
votes
0answers
66 views

Change in behaviour of Series from 11.1

In 11.0, Series[(1 - Sqrt[1-4z])/2, {z, 1/4, 1}] gives 1/2 - I Sqrt[z-1/4] + O[z-1/4]^(3/2) as one would expect. From 11.1 ...
1
vote
0answers
79 views

Series with Dedekind eta function does not expand in 11.3

The following expansion works correctly under versions 7, 10.2, 11.0 ...
0
votes
2answers
46 views

Finding the position of exponents with non-zero coefficients

Suppose that an expansion can be obtained from ...
3
votes
1answer
113 views

About two point Taylor series expansion

From a comment to a two years old question of mine on MSE, I heard (for first time, I must confess) about two point Taylor series expansions; the commenter gave a link to a "quite" recent paper. ...
2
votes
0answers
62 views

Bug using Series on MeijerG[{{}, {}}, {{0, 1/2, 1}, {}}, y] for large y

In Mathematica 11.2.0 on macOS, $Assumptions = {y > 0} Series[MeijerG[{{}, {}}, {{0, 1/2, 1}, {}}, y], {y, Infinity, 0}] reports a result a factor of 2 too ...
1
vote
2answers
154 views

How can I invert this series?

I have a series which is expanded around $x=0$: $$S=2 \sum_{n=0}^4 x^n (c_n\ln x+b_n)$$ where $ c_n=-\frac{1}{2}\left(\frac{(2n-1)!!}{2n!!}\right)^2$ and $b_n=-c_n\left(4 \ln 2+2\sum_{k=1}^n(\frac{1}...
4
votes
1answer
213 views

When to trust Series

I want to evaluate this function near x=1: ...
4
votes
4answers
397 views

Trying to get a Laurent expansion of a symbolic function

Im trying to find the Laurent expansion of the function $$f(z):=\frac{a-b}{(z-a)(z-b)},\quad\text{for }0<|a|<|b|$$ around $z=0$ in the annulus defined by $A:=\{z\in\Bbb C:|a|<|z|<|b|\}$. ...
1
vote
2answers
149 views

Series expansion gives different result

I would like to know the leading order of $x$ in the expression $\sqrt{c+x^2-\sqrt{c^2+x^2}}$, where $x>0$ and $c\in \mathbb{R}$. I tried ...
1
vote
2answers
245 views

Series expansions with negative powers

Here is a minimal example of my problem. I expand a function, f[x], in a Taylor series around a point a == 1/b. The variable <...
5
votes
3answers
279 views

Guess special function from few power series coefficients

I remember having come across a nifty function that guesses some special functions taking some specific values for its parameters, such that when expanded in a power series, its coefficients match a ...
0
votes
1answer
49 views

Series expansion using complex variables

Suppose I have a function F[r_]:= A/r + B/r^2 + Conjugate[C]/r^3 with A, B, C being complex numbers and r being a real variable. I would like to series expand ...
2
votes
2answers
134 views

Solving a second order non-linear D.E using series

I have been trying to solve the following second order non-linear differential equation. $$\frac{\sin ^2(q(z)) \left(3 \left(z^2 q'(z)^2+1\right) \cos (q(z))+z \sin (q(z)) \left(-z q''(z)+4 z^2 q'(z)...
1
vote
1answer
51 views

About the Coefficient command after double Taylor Series

First, let me give the code. These are the matrices. ...
2
votes
2answers
202 views

Asymptotic solution of the saddle-point equation

As an example we have the following equation: $$\sum _{j=1}^{\infty } \frac{r^j}{\left(1-r^j\right)^2}=n$$ Sum[r^j/(1 - r^j)^2, {j, 1, Infinity}] == n I'm ...
3
votes
1answer
62 views

Expanding in small but non zero quantity

I am trying to deal with a function which diverges at $r=0$. I encountered the following function $$f(r)=\int_0^1\frac{x^2(2-x)}{1-x+rx^2}\mathrm dx$$ The paper says that in the limit $r\ll1$, it ...
2
votes
1answer
119 views

Problem with SeriesData

Problem: I need to find the leading order term in an expansion whose leading order behavior is a priori unknown. I can of course go with Series and try different orders, say ...
1
vote
1answer
57 views

Making a notebook with Taylor series of the square root of a determinant more efficient

I have a notebook running for more than five hours. The question is if and how I can make it faster. If there is not a way please leave a comment, so that I know. In the matrices with a final index ...
0
votes
1answer
89 views

Expand terms of series in denominator

I have equation: $a+bx+cx^2+dx^3= \dfrac{1}{f+gx+hx^2+ix^3} \dfrac{\partial}{\partial r}(ax+bx^2+cx^3+dx^4) +\dfrac{1}{(f+gx+hx^2+ix^3)^2} \dfrac{\partial^2}{\partial r^2}(ax^2+bx^3+cx^4+dx^5)$ ...
0
votes
0answers
28 views

Series of Imaginary part of a function

I want to expand Imaginary part of the below function, knowing that all parameters are real. I did like this: ...
4
votes
2answers
102 views

How to get series coefficients from functional equations?

$p(z),t(z)$ are two mutually defined functional equations, while $\widehat{G}(z)$ is the exponential generation function of A182173 (maybe, I am not sure...lol): $$\begin{cases} p(z)=e^{t(z)}-t(z)+2 ...
3
votes
1answer
264 views

Solving a mathematical series inequality with mathematica

I have to show that a_n >= b_n for N{0}, Knowing that a > 0 and b > 0. See the picture How can I show the inequality with mathematica? Here is what I tried, but it didn't work (I began Mathematica ...
2
votes
2answers
285 views

How can I produce Lagrange inversion coefficients?

I am new to Mathematica. I want to produce the Lagrange inversion coefficients by using Bell polynomials. In the following image you can see expressions for the first seven coefficients $A_1$ to $A_7$...
1
vote
2answers
188 views

Special case of hypergeometric function

Trying to figure out if the double sum $$ f(x,y)=\sum_{n=0}^{\infty}\sum_{m=n}^{\infty}\dfrac{(a)_n\,(b)_n\,(1)_m}{(a)_m\,(b)_m}\dfrac{x^n\, y^m}{n!\,m!}, \;\; \text{where} \;\; (z)_n=z(z+1)\cdots(z+n-...
2
votes
1answer
104 views

Multiplying into an infinite sum

Suppose I define y = Sum[Subscript[a, k] x^k, {k, 0, Infinity}] which is a power series representation for y. Now if I multiply <...
0
votes
1answer
46 views

Matrix in exponent [closed]

If, for example, matrix A given as $A = \begin{pmatrix} 0 & x & z \\ 0 & 0 & y \\ 0 & 0 & 0 \end{pmatrix}$ By power series expansion one can obtain $e^A = \begin{...
0
votes
0answers
92 views

Expanding q-series having q-Pochhammer symbols in the summands as a power series [duplicate]

I have Mathematica 11.0. I would like to see power series coefficients of a q-series (having q-Pochhammer symbols in the summands) together in order to look for certain arithmetic information. ...
1
vote
2answers
528 views

How to solve for all coefficients in a sum

The statement of the problem: In the following formula, $$g(u,v) - \sum_{\Delta,l} c_{\Delta,l} u^{\frac{1}{2}} G^{(l)}\Bigg(\frac{1}{2} (\Delta-l),\frac{1}{2} (\Delta-l),\Delta,u,v \Bigg) = 0$$ ...
1
vote
0answers
104 views

Abstract Function of a Function [closed]

I am trying to write a function $V(x) = \sum_{i=1}^{N} p^iV_i(x)$ in that abstract form. The idea is eventually I need to plug this in for a Taylor Expansion of terms of $\ln\left(\frac{\partial V}{\...
1
vote
0answers
60 views

A possible bug of Mathematica

There seems to be a bug for Mathematica when evaluating series expansion for expressions containing log functions. For example: ...
3
votes
3answers
160 views

Simplifying with a Taylor series when $n$ is large, but much smaller than infinity

A calculation yields the following result: $\text{kh}\to \frac{2 \sqrt{3} \sqrt{\nu +n-1} \sqrt{\text{R}^2 n^3+3 \nu +12 \nu n+15 n-3}+6 \nu -6 (2 \nu +1) n-6}{\text{R} (\nu +1) n^2}$ Knowing that $...
1
vote
0answers
109 views

Sum involving hypergeometric function $\mbox{}_1 F_2$

Trying to simplify the sum $$ \sum\limits_{n=0}^{\infty}\dfrac{z_1^n}{n!} {}_{1}F_{2}(1;a+n,1-a+n;z_2), $$ where $a\in(0,1)$, $z_1,z_2>0$, and ${}_{1}F_{2}$ denotes the appropriate version of the ...
6
votes
2answers
208 views

Series expansion gives incorrect result

Bug introduced after 10.4 and persisting through 11.3.0 Mathematica 11.1.1.0 tells me that ...
2
votes
1answer
94 views

Working with Limit on a Root expression

I am trying to diagonalize a somewhat larger matrix that is dependent on several parameters. Applying, say, Eigenvalues to that matrix is quite straightforward, but ...
5
votes
1answer
224 views

How can I get the minimum error term when manipulating Taylor series?

While working on this problem I decided to check some of my work with Mathematica: ln[4]:= Series[u[x + h], {h, 0, 4}] $u(x)+h u'(x)+\frac{1}{2} h^2 u''(x)+\...
1
vote
0answers
149 views

Series expansion of expressions with Log and PolyLog functions (again)

I think that I discovered another (c.f. Series expansion of expressions with Log and PolyLog functions) issue related to Series and ...
1
vote
0answers
47 views

Cannot obtain a limit though the answer can be seen from series expansion

I am considering the order of magnitude of $l_a(s)$ when $s\to\infty$: ...
0
votes
0answers
34 views

Function for putting together a series

For instance Series[Exp[x],...] gives you expansion of Exp[x]. But how would I obtain Exp[x] if I give a summation function?
0
votes
1answer
115 views

Fourier transform of $1/\sin(\pi x)$ - a quest to find the sign function!

While performing the following (presumably) correct manipulations in Mathematica, I obtain a result that is missing a sign function. Is there a mistake in my code, or is there some bug in Mathematica? ...
1
vote
0answers
54 views

Summation with skipping a term

how do i sum the following in Mathematica: the m-th term is 1/(m^2 - n^2), the sum is over odd m and m <> n. i know the answer is -1/(4*n^2) or something like that. thanks !!
0
votes
1answer
203 views

Using Mathematica to expand a taylor series to prove the order of a Runge-Kutta method

Given the Runge-Kutta method, $$w_{i+1}=w_i+\frac{1}{4}k_1 +\frac{3}{8}k_2+\frac{3}{8}k_3 $$ where \begin{align} k_1 &= hf(t_i,w_i)\\ k_2 &= hf(t_i+\frac{2}{3}h,w_i+\frac{2}{3}k_1)\\ k_3 &=...
6
votes
2answers
197 views

Problem with `Derivative`/`Series` and `InverseFunction`

Here is the Mathematica code I'm asking about: ...
5
votes
2answers
93 views

Possible bug in integral with PolyLog and Zeta

$Version "10.1.0 for Microsoft Windows (64-bit) (March 24, 2015)" The following integral was nicely and quickly done by Mathematica ...
1
vote
1answer
396 views

Plotting a few Maclaurin Series in Mathematica

I would like to plot the Taylor polynomials for several functions. Specifically: Exp[Sin[x]] ( x^2 + Exp[ x ] )^( -1 ) and ...
2
votes
1answer
796 views

Finding power series solution for differential equation in Mathematica [duplicate]

I know this topic has been covered before, but I've tried all the solutions I can find from other users' questions and none of them have worked. I need to find a power series solution to the ...
1
vote
2answers
255 views

Taylor expansion in terms of small parameters of a polynomial fraction

I have some fractional expression like $$ \frac{x^2+xyz+z^2}{x^2-yz+x^2z} $$ and I know that $x \ll y \ll z$. Really I want to divide through by $z^2$ and then take a Taylor series expansion in ...

1 2 3 4 5 13