Questions tagged [series-expansion]
Questions on dealing with series data and constructing power series expansions of functions.
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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:
$$
\begin{equation}
...
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Series expansion with nth term of $\left(\frac{\sin (x)}{x}\right)^a$ [migrated]
Using Mathematica, I need an expansion with $n$th term of $$ 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}$$ then I need a ...
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Approximating exponential generating function (EGF) from values of generating function (OGF)
I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
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Zassenhaus formula in Mathematica
I'm looking for Mathematica implementation of Zassenhaus formula -- given two matrices $A,B$, truncate the expansion of $\exp(A+B)$ from this paper:
$$e^{t(A+B)}= e^{tA}e^{tB}\prod_{n=2}^\infty e^{t^n ...
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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}}-\...
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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-(...
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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 ...
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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 ...
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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}$ ...
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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 ...
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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 ...
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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 ...
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SeriesCoefficient stops working on EllipticTheta in v13.2
In v12, the following SeriesCoefficient computation gives the expected result,
...
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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 &...
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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\...
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Speed up a infinity series
Is there any trick to speed up the plotting of my function u[x,t]?
...
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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
...
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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 ...
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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\...
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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)...
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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 ...
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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 (<...
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Couldn't compute the coefficients of this series
Mathematica doesn't expand the series of this function (I'm using Wolfram Cloud version)
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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 ...
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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:
...
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'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}=\...
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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\...
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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 ...
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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
...
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Unable to expand this solution in the form of Hypergeometric function
I want to write this solution into hyper geometric function
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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 ...
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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
...
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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 ...
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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 ...
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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 ...
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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$ ...
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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 ...
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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:
...
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Extracting a logarithmic divergence of an expression using Series
Consider the following expression:
...
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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:
...
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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. ...
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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,$$
in the sense of "asymptotics", the equal sign is ...
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Plotting contours of a two-variable function containing a sum
I'm trying to use Mathematica to plot contours of a rather intricate two-variable function. The equation describes the velocity profile for laminar flow in a tube of rectangular section, namely:
$$
{w^...
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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 ...
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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 ...
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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:
...
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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 ...
