Questions tagged [series-expansion]
Questions on dealing with series data and constructing power series expansions of functions.
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The Baselproblem from Euler
$Zeta(2) = 1/6 Pi^2$
This is the value of the Riemann Zeta function "for the number 2
Euler the famous mathematician first calculated this symbolic value in the 18 century in his "Basel ...
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Convergent Taylor series unrecognized by Sum
I am trying to understand why Sum does not recognize a particular Taylor series as convergent. I have defined a function 'series' like this, that computes the Taylor series of a function 'f' at 'x', ...
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How to find the asymptotic envelope of a function?
Context
I am interested in the asymptotic behaviour of the envelope of a given function.
Unless I missed it, it would be of interest to have a Mathematica function which
when given
...
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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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How to write code for SeriesCoefficient to work for non integral coefficients?
I have a function of $r$ which I expand at $\infty$ using Series. It is a complicated and messy function, with a parameter $0 \leq \epsilon < 1$. After expansion,...
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How to access the Wolfram Data Repository
How I can obtain the raw data of different epidemics in different viruses such as COVID-19, ebola, influenza, etc.?
I consulted the site but I can't figure out how to extract the data in real time to ...
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Set the value of a parameter in a Series expression [closed]
I have a lengthy expression resulting from a series expansion in some dummy variable $e$ which I now wish to set equal to 1. However, when I try to use a ReplaceAll ...
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Asymptotic integral expansion at infinity [closed]
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
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How can I make sure that when I write in the wolfram language, I'm writing exactly what I intend (or what others are writing)? [closed]
I'm someone who is just starting to do some hobbyist math on my own, using various tools like wolfram alpha, and I have a question about the wolfram command line. My question is as follows
I'm not ...
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Finding constant term in product expression
I've an expression which is product of 20 or more factors of polynomial, something like $$\left(1-\frac{pq}{z^i}\right)(1+pq z^j+z^k)$$ and I want to find coefficient of $z^0$. SeriesCoefficient works ...
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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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Perturbing a tensorial expression
I am new to Mathematica. I am trying to simplify an expression of the some form like:
$$
n_i \sigma_{ij} n_j - \gamma n_i \hat{\sigma_{ij}} n_j = 2 + v_i x_i + \kappa E_{ij} \chi_{ji}
$$
There are ...
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Can Mathematica simplify an ordinary differential equation (ODE) by assuming a power series solution and obtain the recurrence relation?
I have an ordinary differential equation and want to solve it using power series as ψ[x_] = Sum[Subscript[a, n] x^n, {n, 0, ∞}] to obtain the recurrence relation ...
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Truncation by coefficient size
I have a series with e.g. Chebyshevs: $\sum_i^N a_i T_i(x)$
where they are decreasing in size with increasing $i$.
So now suppose I multiply two such series: $(\sum_i^N a_i T_i(x))(\sum_i^N b_i T_i(x))...
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Strange behavior of error function series expansion at infinity
Evaluating
Normal[Series[1/ (1 + xi Erf[xi]), {xi, ∞, 2}]]
Normal[Series[1/( xi (1/xi + Erf[xi])), {xi, ∞, 2}]]
gives
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Proper treatment of roots and powers in Series?
I have the following problem in Mathematica 9 on Linux. I let Mathematica compute the Series expansion:
...
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BUG: Why is Series[] getting this expression wrong?
EDIT: Wolfram confirmed this is a bug in Series[], and they're looking into it.
I'm trying to generate a 2nd-order Taylor series in theta for a complicated expression "mdel" using Series[...
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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 ...
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Using Integrate and then Series seem to produce a wrong result
Bug introduced in 11.1.0.0 or earlier and persisting through 14.0 or later
Run this:
...
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Limit of Hypergeometric Functions
Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following
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Series behavior for self-defined function
How to make Series give the correct expansion of a self-defined function?
For example, for some reason, I use f[x] to represent ...
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Asymptotic solution of a system of ODEs
I have the following system of Ordinary Differential Equations (ODEs) together with initial values
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Expanding polynomials using valuation
I would like to expand the polynomial $p(\lambda) = \sum_{i=0}^{d} a_{i} \lambda^{i}$, as $F(p(\lambda), \lambda_{0})= \min_{j} [ val(a_{j}) + j \lambda_{0} ] $ with $\lambda_{0}$ being a real ...
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Evaluating series expansion is very slow
I need in my work to get series expansion of
(2 E^x x HypergeometricPFQ[{1}, {1/2 + E^-x/4, 1 + E^-x/4}, -(x^2/4)])/Gamma[E^-x/2] + x^(1 - E^-x/2) Sin[x]
up to $n=...
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Replacing function by another one each time it appears
I have expression involving Cos of some parameters. I would like to replace those Cos by their infinite series each time they do appear.
I tried the following which doesn't work:
...
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Why does Mathematica get the infinite sum of this representation of the taylor series of $\cos^2(x)$ wrong?
Bug introduced in 12.0 or earlier, persisting through 13.2 or later
Mathematica correctly identifies this sum as $\cos(x)$:
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Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
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First argument -h is not a valid variable
I am trying to simpligy my life by defining a function that gets the series expansion of $f(x+h)$ and $f(x-h)$ (at $x=0$) by the following code,
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Limit giving indeterminate result
I have a function $r_h(v)$ given by,
$$r_h (v) = \left(\frac{m_0}{2 g} e^{-g v_f} \left(-1+e^{-g(v - v_f)} \right) \right)^{1/5}$$
where $m_0$ and $g$ are just numbers. I want to take the limits of ...
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How to solve recurrence equation using RSolve?
How can I solve the following recurrence equation while I dont have the initial values?
Is it possible to solve this using RSolve?
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Series expansion message with special functions
I am trying to expand the following expression in alp,eps. Since the expression involves Hypergeometric2F1, I am using HypExp.
<...
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Finding the coefficients of a decomposition of complicated expression into products of special functions
I have the following expression which arises while studying minimal models in CFT. The 4-point amplitude of scalars is given by
$$G(z,\overline{z}) = ((1-z)(1-\overline{z}))^{\frac{m-1}{m+1}}\mathcal{...
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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 ...
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An apparent error with Chebyshev polynomials
I am on 11.0.1.0
SeriesCoefficient[ChebyshevU[n,x],{x,0,m}] returns
...
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Is it possible to circumvent a bug inside SeriesCoefficient?
Bug introduced in 9.0.1 or earlier and fixed in 13.3.
As far as I can tell, there seems to be a bug in SeriesCoefficient:
...
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Why do I get a wrong result from SeriesCoefficient?
Bug introduced in 7.0.1 or earlier and fixed in 13.3
Consider the following code:
func[x_] = Sin[x^3]/(x - 1/3);
c[n_] = SeriesCoefficient[func[x], {x, 0, n}]
<...
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Proving an expression from Mathematica which is clearly visible from Plots
I have the following Mathematica code:
...
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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 ...
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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 ...
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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 ...
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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 ...
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Laurent series expansion
Can someone share how to find the Laurent series expansion of
$$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$
centered at $0$ on the annulus $1<|z|<2$?
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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 ...
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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:
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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 ...
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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 ...
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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 ...
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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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