21 questions linked to/from Chebyshev Approximation
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### The coefficients with NIntegrate [duplicate]

I try to compute the coefficients b[i] , $i=2,...,m$. I got the following output: ...
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### About multi-root search in Mathematica for transcendental equations

I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range? Perhaps ...
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### How can I differentiate numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
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### How to solve ODE with boundary at infinity

y''[x] - x y[x] == 0 y[0] == AiryAi[0], y[∞] == 0 The analytic solution to this ODE is the Airy function y[x] == AiryAi[x] if ...
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### Determining which rule NIntegrate selects automatically

I need to numerically integrate a highly oscillatory function over the semi-infinite domain $(0,\infty)$: $$\int_0^\infty \frac{\sin^2(x) \sin^2(1000 x)}{x^{5/2}}\mathrm dx$$ Since the Levin rule (...
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### What's inside InterpolatingFunction[{{1., 4.}}, <>]?

I'm curious what's inside the InterpolationFunction object? For example: ...
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### Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
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### Interpolating data with a step

Suppose I have some data with a step in it: data = {{1, 1}, {2, 2}, {3, 3}, {3, 4}, {4, 5}, {5, 6}}; Interpolation will complain about this and not give you an ...
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### Convert polynomial to Chebyshev

I want to convert a polynomial in "standard form" to Chebyshev form. Here's one way to do it: ...
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### How can I find the inverse of an interpolating function?

I am numerically solving the coupled differential equations (for $t[\tau]$ and $r[\tau]$) below. Mathematica outputs two interpolating functions as solutions. I would like to invert one of these ...
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### Analytical approximation of indefinite integral on a given interval to a given precision

I'm looking for an analytical approximation of $\int_a^b e^{-x^2}\mathrm{erf}(x+c) dx$ that would be accurate to precision $\varepsilon$ for $a,b,c$ within a certain range. How do I ask Mathematica ...
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### FunctionInterpolation over an open interval

I'm trying to obtain an arc-length parametrization of a spline curve as per https://mathematica.stackexchange.com/a/8456. To do so, I need to calculate partial derivatives of such curve and then ...
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### Precompiling a Whittaker function

Is there a way to speed up the evaluation of special functions in Mathematica? I am particularly interested in the Whittaker W function. For instance, the following piece of code: ...
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### Find all roots of a function with parabolic cylinder functions in a range of the variable

I want to find all roots of a function involving Parabolic Cylinder Functions. In what follows, I define 2 variables $\xi1$ and $\xi2$, which in turn depend on $\omega$. My function is then defined as ...
I'm trying to find a polynomial representation for this horrendous function: f(x)=\frac{\frac{1}{2}-\frac{4}{1+10,000x}\sum_{n=0}^{\infty}\frac{(-1)^n}{\big[\big(n+\frac{1}{2}\big)\pi\big]^4}\tanh\...