Questions tagged [series-expansion]
Questions on dealing with series data and constructing power series expansions of functions.
877
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Why does Mathematica get the infinite sum of this representation of the taylor series of $\cos^2(x)$ wrong?
Mathematica correctly identifies this sum as $\cos(x)$:
Sum[((-1)^n x^(2 n))/(2 n)!, {n, 0, Infinity}]
Mathematica also correctly identifies this product of sums ...
2
votes
3
answers
140
views
Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
- 123
3
votes
2
answers
161
views
First argument -h is not a valid variable
I am trying to simpligy my life by defining a function that gets the series expansion of $f(x+h)$ and $f(x-h)$ (at $x=0$) by the following code,
...
- 33
0
votes
0
answers
46
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Limit giving indeterminate result
I have a function $r_h(v)$ given by,
$$r_h (v) = \left(\frac{m_0}{2 g} e^{-g v_f} \left(-1+e^{-g(v - v_f)} \right) \right)^{1/5}$$
where $m_0$ and $g$ are just numbers. I want to take the limits of ...
- 661
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votes
0
answers
96
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How to solve recurrence equation using RSolve?
How can I solve the following recurrence equation while I dont have the initial values?
Is it possible to solve this using RSolve?
...
1
vote
0
answers
78
views
Series expansion message with special functions
I am trying to expand the following expression in alp,eps. Since the expression involves Hypergeometric2F1, I am using HypExp.
<...
- 1,604
1
vote
0
answers
25
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Finding the coefficients of a decomposition of complicated expression into products of special functions
I have the following expression which arises while studying minimal models in CFT. The 4-point amplitude of scalars is given by
$$G(z,\overline{z}) = ((1-z)(1-\overline{z}))^{\frac{m-1}{m+1}}\mathcal{...
- 185
2
votes
1
answer
73
views
An apparent error with Chebyshev polynomials
I am on 11.0.1.0
SeriesCoefficient[ChebyshevU[n,x],{x,0,m}] returns
...
- 1,874
1
vote
1
answer
82
views
Proving an expression from Mathematica which is clearly visible from Plots
I have the following Mathematica code:
...
- 899
1
vote
0
answers
85
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Trying to use Linear Optimization to solve inequalities
I am trying to use Mathematica to get bounds on some quantities using a certain procedure which is called conformal bootstrap. For the users who don't come with a background in HEP, I am basically ...
- 185
2
votes
1
answer
102
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How to convert DifferenceRoot into a special function?
Clear["Global`*"];
f[z_] := z^(2 m) /(1+z)^m
res = SeriesCoefficient[f[z], {z, -1, -1},
Assumptions -> Element[m, PositiveIntegers]]
The result ...
- 1,943
1
vote
1
answer
100
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Series expansion of Beta function in Mathematica
How one should do the series expansion of Beta function $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ around an arbitrary negative integer let's say $x=-k$ and $y=-l$? I want a symbolic expression ...
- 39
1
vote
1
answer
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A request on the kernel of the function ”SpheroidalEigenvalue“ in Mathemitica 11
I want to understand the algorithm of the function "SpheroidalEigenvlaue" in Mathematica 11, especially the code of the following command.
Then I will try to use this algorithm to reproduce ...
- 31
4
votes
1
answer
110
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A simple series expansion which seems to be wrong
Trying to answer this question, I made the following input
FullSimplify[SeriesCoefficient[ArcTanh[x]/(1 + x^2), {x, 0, n}]]
I shall not type the results but,
not ...
- 731
0
votes
1
answer
71
views
Series expansion for expression with parameter?
I would like to compute the following expansion.
Series[(A + p/x^a)^2, {x, 0, 1}]
where $a>0$. However Mathematica simply returns the expression, unless I ...
- 307
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votes
0
answers
115
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How to solve or test the interval of Uniform Convergence of function series?
How to solve or test the interval of Uniform Convergence of function series? (ref2)
e.g.
$\sum_{k=1}^{\infty}\left(z^{k-1}-z^k\right)$
The convergence interval of this series can be got by ...
- 1,943
1
vote
1
answer
215
views
Discrepancy with Hurwitz Zeta function
I've come across an issue while using Wolfram Mathematica that I don't quite understand.
Consider the following symbolic computation:
...
- 27
1
vote
1
answer
45
views
How to obtain a list of pairs of exponents in a double series expansion?
Let's say we have a function of two variables $f(x,y)$ and we work out its Taylor expansion up to some power. I would like to use Mathematica to construct a list of all exponents that appear in the ...
- 481
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votes
0
answers
185
views
How to approximate an exponential series?
Consider the following expression
$$
y_j= \sum_{k=0}^{L} \frac{e^{-\sum_{i=-k}^k(k-|i|)x_{j+i}}-e^{-\sum_{i=-k}^k(k+1-|i|)x_{j+i}}}{\sum_{i=-k}^k x_{j+i}}\tag{1}
$$
for $1\leq j \leq L$. Given smooth ...
- 4,147
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1
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80
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Comparing two power series and extracting their coefficients
I have a standard mathematical problem that I was wondering how to solve efficiently by mathematica. Here is the problem.
I have two power series expansions of a function ...
- 417
0
votes
0
answers
32
views
Weird expression for function Series-Expansion with Gamma function for different values of gamma coefficient
I extract the function jin[r] by solving eqsynin, and then I develop the function's series (around zero) to generate an equation for m1in and m2in based on esyn and gamma, knowing that the function ...
1
vote
1
answer
117
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How to expand $\frac{1}{(1-z-z^2)}$ into a power series [closed]
How can I get Mathematica to expand
$$\frac{1}{(1-z-z^2)}$$
into a power series so that I can pick out the coefficients.
- 197
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0
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How to accelerate Inverse[] for positive definite matrices symbolically?
I am trying to construct a positive definite matrix based on the multiquadric radial basis function (RBF) for a set of thirteen points symbolically in order to later approximate the Laplacian operator ...
- 1,843
4
votes
1
answer
116
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Why `AsymptoticSolve` doesn't work for a multivariate implicit function?
I started by defining
...
- 163
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0
answers
26
views
Attempt to evaluate a series returning Indeterminate while running Plot
I have a function in the form of a Series from a prior calculation:
sol2D = SeriesData[a, 0, {Rational[3, 16] Pi,
Rational[-5, 2], Rational[3, 4] Pi}, 0, 3, 2]
...
1
vote
0
answers
95
views
Taylor series loop
I'm a beginner not only in Mathematica but also in programming in general, and so I'm not really sure where my problem lies exactly and I'd be glad to receive any guidance.
Using the Taylor series for ...
2
votes
1
answer
81
views
Why does Series give two different results for given function?
I want to do Series of this function x^12 Cos[y x]^2 Sin[y x]^6 Sin[(-1 + Sqrt[3]) y x]^2 around ...
- 349
2
votes
1
answer
107
views
Making Series Solutions Look Nicer
My students and I are using AsympototicDSolveValue[] to find power series solutions to linear differential equations at 0. For example, the following code gives me a solution up to degree 7.
I'm ...
- 5,409
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1
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119
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Can Mathematica estimate this complex function?
Mathematica has given me a function in $x,r$ given by
...
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2
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1
answer
145
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How to obtain the Taylor expansion of any function? [duplicate]
How to obtain the Taylor expansion of any function?
Like the Taylor expansion of any function in the picture. How can I obtain the Taylor expansion of any function if I input it?
...
- 2,879
0
votes
0
answers
23
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Different results for same output
In a power series solution method, I am trying to find the roots of an equation. By changing the parameters I need to get the roots. The problem is the parameter value is provided as 0.15 which gives ...
4
votes
4
answers
209
views
Solving PDE with power series
I would like to solve the PDE
$$\partial_{x}f(x,y) + f(x,y)^2 = g(x,y)$$
with $f(0,0)=0$ and $\partial_y f(0,0)=0$ using a power series ansatz, i.e. I have an explicit expression for $g(x,y)=\sin(x+y)\...
0
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0
answers
69
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Approximating Exp[-x] in partial fraction form [duplicate]
I'm looking to obtain order-$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:
$$
\begin{equation}
...
- 8,041
3
votes
1
answer
64
views
Approximating exponential generating function (EGF) from values of generating function (OGF)
I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
- 8,041
4
votes
1
answer
335
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Zassenhaus formula in Mathematica
I'm looking for Mathematica implementation of Zassenhaus formula -- given two matrices $A,B$, truncate the expansion of $\exp(A+B)$ from this paper:
$$e^{t(A+B)}= e^{tA}e^{tB}\prod_{n=2}^\infty e^{t^n ...
- 8,041
2
votes
2
answers
208
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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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votes
1
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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-(...
0
votes
0
answers
42
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
82
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 ...
- 259
1
vote
2
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90
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}$ ...
- 165
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votes
3
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108
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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 ...
- 137k
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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 ...
- 2,879
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0
answers
75
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SeriesCoefficient stops working on EllipticTheta in v13.2
In v12, the following SeriesCoefficient computation gives the expected result,
...
- 513
4
votes
0
answers
134
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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 &...
- 591
4
votes
2
answers
131
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\...
- 731
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vote
1
answer
134
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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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2
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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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2
answers
143
views
0
votes
0
answers
88
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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 ...
- 899
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2
answers
136
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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\...
