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Tanh-sinh quadrature is a method for numerical integration introduced by Hidetoshi Takahashi and Masatake Mori in 1974. [1] It is especially applied where singularities or infinite derivatives exist at one or both endpoints. The method uses hyperbolic functions in the change of variables
The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the polynomial of degree 3 ( y ( x ) = 7 x 3 – 8 x 2 – 3 x + 3 ), the 2-point Gaussian quadrature rule even returns an exact result.
SciPy (pronounced / ˈ s aɪ p aɪ / "sigh pie" [2]) is a free and open-source Python library used for scientific computing and technical computing. [3]SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and engineering.
The following Python code with the SymPy library will allow for calculation of the values of and to 20 digits of precision: from sympy import * def lag_weights_roots ( n ): x = Symbol ( "x" ) roots = Poly ( laguerre ( n , x )) . all_roots () x_i = [ rt . evalf ( 20 ) for rt in roots ] w_i = [( rt / (( n + 1 ) * laguerre ( n + 1 , rt )) ** 2 ...
Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. [1] The window function means that the signal near the time being analyzed will have higher weight.
The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C ∞ of smooth functions) only if is a positive, even integer. Otherwise, the function has ⌊ β ⌋ {\displaystyle \textstyle \lfloor \beta \rfloor } continuous derivatives.