Questions tagged [series-expansion]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
57 views

Approximating Exp[-x] in partial fraction form

I'm looking to obtain order-$k$ approximation of $\exp(-z)$ for real-valued $z$. $$R_k(z)\approx \exp(-z)$$ The constraint is that I need the result in partial fraction form, ie: $$ \begin{equation} ...
0 votes
0 answers
65 views

Series expansion with nth term of $\left(\frac{\sin (x)}{x}\right)^a$ [migrated]

Using Mathematica, I need an expansion with $n$th term of $$ f(x)=\left(\frac{\sin (x)}{x}\right)^a $$ about $x=0$ where $a\geq 0$ or if $$f(x)=\sum_{n=0}^{\infty} b_{2n }x^{2n}$$ then I need a ...
  • 75
3 votes
1 answer
51 views

Approximating exponential generating function (EGF) from values of generating function (OGF)

I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
4 votes
1 answer
293 views

Zassenhaus formula in Mathematica

I'm looking for Mathematica implementation of Zassenhaus formula -- given two matrices $A,B$, truncate the expansion of $\exp(A+B)$ from this paper: $$e^{t(A+B)}= e^{tA}e^{tB}\prod_{n=2}^\infty e^{t^n ...
2 votes
2 answers
112 views

Series expansion using binomial theorem in Mathematica

The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by $$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
  • 159
0 votes
1 answer
38 views

Solving series solution of differential equation

AsymptoticDSolveValue[2x*y''[x] -(3+2x)*y[x] +1 == 0, y[x], {x, 0, 5}], this differential equation command, is not outputting the correct solution. The solution should be like y=C1(1+(1/3)x-(1/6)x^2-(...
0 votes
0 answers
31 views

Solving differential equation to series solution

I tried to solve y''+x^2y=0 this differential equation to the series solution, so I put the command of AsymptoticDSolveValue[y''[x]+x^2*y[x]==0, y[x], {x,0,5}] like this, but the output shows like ...
0 votes
1 answer
47 views

Expansion of standard inverse normal cdf

Let $\Phi: x\rightarrow y$ be the CDF of a standard normal distribution. The range of $\Phi(x)$ is $[0,1]$ and the domain is all real numbers. I want to get a series expansion of $\Phi^{-1}(y)$ around ...
1 vote
2 answers
78 views

Series from an integral and output as a function

I have a simple question, I am just stuck on syntax. I want to have a series of function $Z(\lambda)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} d x e^{-\frac{x^2}{2 !}-\frac{\lambda}{4!} x^4}$ ...
0 votes
0 answers
23 views

Why complementary solution of an ode changes when adding term on the RHS?

I found a case where AsymptoticDSolveValue gives correct solution to a second order ode when the RHS is zero. i.e. complementary solution is correct. As we all ...
  • 127k
4 votes
3 answers
102 views

Why earlier terms generated from AsymptoticDSolveValue change when increasing the order?

I was comparing my hand solution with Mathematica's. I noticed the particular solution terms generated for a Frobenius series solution change depending on the order. This should not happen. As when ...
  • 127k
2 votes
1 answer
60 views

Then can the result of the general term formula be written in subsection form?

s[n_] = n^2 - 2 n + 3 RSolve[a[n + 1] == s[n + 1] - s[n], a[n], n] The above example shows that the general term formula of the sequence should be in a piecewise ...
  • 793
1 vote
0 answers
62 views

SeriesCoefficient stops working on EllipticTheta in v13.2

In v12, the following SeriesCoefficient computation gives the expected result, ...
  • 459
4 votes
0 answers
114 views

InverseSeries giving incorrect result

Somehow in Mathematica 13.2.0.0, InverSeries generates incorrect results. Let's look at the following two series that differs from each other by a constant number &...
  • 591
4 votes
2 answers
128 views

Looking for the asymptotics of an asymptotics

I am trying to polish my second answer to this question in Mathematics Stack Exchange. The problem is to find the asymptotics of $t$, solution of the implicit equation $$\color{blue}{\left(1-2 x^2\...
1 vote
1 answer
127 views

Speed up a infinity series

Is there any trick to speed up the plotting of my function u[x,t]? ...
  • 117
3 votes
2 answers
103 views

Is there something faster than AsymptoticDSolveValue for getting the terms in a power series solution?

Here's an example of a differential equation which Mathematica 13.1 just returns without solving ...
  • 315
4 votes
2 answers
139 views

Mathematica flips the sign of a Maclaurin series

I have the following Mathematica code: ...
  • 689
0 votes
0 answers
83 views

Recursion error for a series expansion while using RGTC code

I am trying to use the RGTC code found on this website to calculate the series expansion for the given differential equation. I get a recursion error when I use this code: Recursion depth of 256 ...
  • 689
1 vote
2 answers
135 views

Finding an elementary function growing asymptotically as the integral of a sequential product

I am trying to understand how grows the function $f:\mathbb{N}\to\mathbb{R}$ Integrate[(1-Product[1-Exp[-2*j*t/(k*(k+1))], {j, 1, k}]), {t,0,\[Infinity]}] for $k\to\...
2 votes
2 answers
303 views

Asymptotic integral computation takes too long

I am trying to understand how grows the function $k\mapsto\int_{0}^{\infty} \left({1 - \left(1 - \exp(-t/k)\right)^k}\right)dt$ for $k\to\infty$, and I expect a result asymptotically equal to $k\log(k)...
2 votes
0 answers
31 views

Can we approximate a matrix power series like NSum does?

Essentially, the following does not work, and I'm wondering if it can be made to: NSum[ MatrixPower[B,n], {n,0,∞}] (Here B is a ...
  • 121
0 votes
0 answers
49 views

I need a recurrence relation or function for a series [Solved]

I am not sure if what I need is a function or a recurrence relation to extend what you see below to substantial n. In words: I have a series. Starting with n=3 (<...
  • 672
0 votes
1 answer
48 views

Couldn't compute the coefficients of this series

Mathematica doesn't expand the series of this function (I'm using Wolfram Cloud version) ...
2 votes
0 answers
58 views

Asymptotic expansion for a function containing irrational exponents

I'm trying to find an asymptotic expansion of a function containing irrational exponents in the form of a power series (which also might contain irrational exponents). It works correctly if I request ...
0 votes
1 answer
62 views

Plot Taylor expansion of polynomial

I have a polynomial: f[x_] := x^3 + 2*x^2 + 4 and I create a function that implements the Taylor expansion: ...
  • 321
0 votes
1 answer
162 views

'Series expansion' of abstract matrix expression

Let A is a nilpotent matrix. $\boldsymbol{A}^l=\mathbf{0}$ To derive $(\boldsymbol{I}-\boldsymbol{A})^{-1}=\boldsymbol{?}$ The answer in the textbook is $(\boldsymbol{I}-\boldsymbol{A})^{-1}=\...
  • 1,719
1 vote
1 answer
119 views

Trouble finding inverse of a function

I have the following Mathematica code: ...
  • 689
1 vote
3 answers
189 views

Series expansion of the integral from its numerical values

I have already asked a similar question regarding approximating Taylor series of the function from noisy data. This is another example I am having trouble with. Consider the integral $$I(x)=\int_0^1\...
  • 73
3 votes
1 answer
131 views

How to do the series expansion of terms with PolyLog faster?

I have the following expression that I need to series expand, around t=0 (PolyLog[3, 1 + Tanh[J t]] - PolyLog[3, 1 - Tanh[J t]])Tanh[J t] The amount of time to ...
  • 177
3 votes
1 answer
65 views

Collecting even power terms in Ising problem

I am trying to solve the Ising model for $4 \times 4$ square lattice without a magnetic field. The calculation involved evaluating the expression below, After writing the code as ...
  • 247
0 votes
0 answers
49 views

Unable to expand this solution in the form of Hypergeometric function

I want to write this solution into hyper geometric function ...
0 votes
0 answers
41 views

Taylor series expansion of a challeging type of polynomial with two summation signs

I have a few questions about series expansions of a particular and difficult type of polynomial written in terms of two summations signs. It should be remarked that I have also read Mathematica's ...
  • 159
3 votes
1 answer
131 views

Calculating power series of quantum operators on kets

I am using the "Quantum" add-on package to perform some quantum mechanical calculations using SU(1,1) generators. The code that I've developed reads ...
  • 749
1 vote
1 answer
113 views

Problem in getting coefficients list of power series solution of y''[x]*y'[InverseFunction[y][x]] - y'[x] == 0?

I have tried to get solution of the following ODE: y''[x]*y'[InverseFunction[y][x]] - y'[x] == 0 Using the code given below, the solution is a formal power series ...
7 votes
1 answer
179 views

How to get more terms with the Series[] expansion of InverseErf[x] around x=1?

The inverse error function, which is given by InverseErf[x], is quite important in statistics as it gives the confidence levels around a 1D Gaussian, for example ...
  • 1,357
3 votes
0 answers
129 views

Cannot Understand nth Derivative of x/ArcTan[x]

The nth derivative of x/ArcTan[x]: f[x_, n_] = D[x/ArcTan[x], {x, n}] Evaluates to: I cannot get this general from to return ...
0 votes
0 answers
69 views

How to do this recursion relation in Mathematica effectively?

I have a function $h_{\Delta,l}(r,\eta)$ satisfying \begin{equation} h_{\Delta,l}(r,\eta)=\tilde{h}_{l}(r,\eta)+\sum_{k}\frac{c(k)}{\Delta-(1-l-k)}r^{k}h_{1-l+k,l+k}(r,\eta) \end{equation} where $k$ ...
  • 51
2 votes
1 answer
77 views

Series Expansion of EllipticNomeQ differs from older Mathematica Version

I am trying to follow the numerical approach on how to calculate EllipticE and EllipticK following this paper. In there on ...
  • 4,857
0 votes
0 answers
39 views

Scaling a variable multiplied by power series is not giving expected output

I have a power series getting multiplied by a variable, which gets scaled by the small parameter used in the power series. As an example, in the following dummy series: ...
6 votes
3 answers
141 views

Extracting a logarithmic divergence of an expression using Series

Consider the following expression: ...
  • 1,317
2 votes
1 answer
109 views

Formal variables no longer simplifying well on new Mathematica version?

I work with a lot of formula manipulation, modular forms etc. For example, the Dedekind eta function comes up a lot, and I'm often interested in the following sort of code: ...
  • 1,317
0 votes
0 answers
74 views

Mathematica 13.0 appears to automatically reverse nested series which contain logarithms. Is there a way to prevent this behavior?

When performing manipulations on nested series, I need the series to retain the nesting order I originally defined. In Mathematica versions 8.0 and 10.4, this was almost always the default behavior. ...
  • 195
0 votes
1 answer
158 views

Can Mathematica solve an ode asymptotically as x goes to infinity?

Given the following ode for $x\rightarrow\infty$: $$\left(x f(x)^3 \left(\frac{(x f^\prime)^\prime}{x}\right)^\prime\right)^\prime=0,$$ in the sense of "asymptotics", the equal sign is ...
0 votes
0 answers
54 views

Plotting contours of a two-variable function containing a sum

I'm trying to use Mathematica to plot contours of a rather intricate two-variable function. The equation describes the velocity profile for laminar flow in a tube of rectangular section, namely: $$ {w^...
  • 21
4 votes
1 answer
85 views

Expanding a Matrix Vector product in powers of epsilon

I am attempting a perturbation expansion in Mathematica. As part of this, I would like to expand a matrix-vector product where the vectors are given in powers of epsilon. Eventually, I'd like to ...
  • 79
4 votes
2 answers
182 views

Numerically solving nonlinear PDE $u_t = G(u,u_y,u_{xy},u_{xx}u_{yy})$ with unbounded initial condition

I am trying to numerically solve some nonlinear partial differential equations similar to the example given below, for which I have been unable to obtain stable numerical solutions due to some ...
  • 143
6 votes
3 answers
341 views

Neglecting higher order terms in a Lagrangian

I have a lagrangian which is modified by variable change. I want to neglect all the 4th order and higher terms in the new lagrangian. The code being used is given below: ...
  • 689
0 votes
1 answer
66 views

NonlinearFit with series coefficients

I have a set of data: ...
0 votes
1 answer
152 views

How to plot a graph for the solution to a differential equation

I am very new to Mathematica and am struggling to get a plot for a differential equation I need solving. I am doing a simplified version of the lane-emden equation for n=1 so have the follwoing ...

1
2 3 4 5
17