19
Bernoulli and Euler Polynomials
Abstract
Focus of this chapter are Bernoulli numbers and polynomials, and Euler numbers
and polynomials in the complex domain. For the evaluation several methods
can be used in dependence of the polynomial degree and argument: Direct
integration, direct integration in combination with argument transformations, or
expansions with respect to trigonometric series. Corresponding MATLAB code
can be downloaded.
Focus of this chapter are Bernoulli numbers and polynomials, and Euler numbers
and polynomials. Applications can be found, e.g., in statistical physics.
Function Overview
Bernoulli The SPECFUNPHYS class bernn returns the Bernoulli numbers and
the SPECFUNPHYS class bernpoly the corresponding polynomial coefficients.
The SPECFUNPHYS class bernx allows the evaluation of Bernoulli polynomials. The computation is not restricted to the interval [0, 1], but complex numbers
are only supported for the polynomial representation.
Euler The SPECFUNPHYS class eulen returns the Euler numbers and the
SPECFUNPHYS class eulepoly the corresponding polynomial coefficients.
The SPECFUNPHYS class eulex supports the direct evaluation of Euler polynomials. The computation is not restricted to the interval [0, 1], but complex
numbers are only supported for the polynomial representation.
Methods The SPECFUNPHYS class bepolymeth comes with additional methods (plot and computing the n-th derivative) for the classes bernpoly and
eulepoly, and bexplot with an additional plot-method for bernx and
eulex.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_19
225
Bernoulli and Euler Polynomials
Abstract
Focus of this chapter are Bernoulli numbers and polynomials, and Euler numbers
and polynomials in the complex domain. For the evaluation several methods
can be used in dependence of the polynomial degree and argument: Direct
integration, direct integration in combination with argument transformations, or
expansions with respect to trigonometric series. Corresponding MATLAB code
can be downloaded.
Focus of this chapter are Bernoulli numbers and polynomials, and Euler numbers
and polynomials. Applications can be found, e.g., in statistical physics.
Function Overview
Bernoulli The SPECFUNPHYS class bernn returns the Bernoulli numbers and
the SPECFUNPHYS class bernpoly the corresponding polynomial coefficients.
The SPECFUNPHYS class bernx allows the evaluation of Bernoulli polynomials. The computation is not restricted to the interval [0, 1], but complex numbers
are only supported for the polynomial representation.
Euler The SPECFUNPHYS class eulen returns the Euler numbers and the
SPECFUNPHYS class eulepoly the corresponding polynomial coefficients.
The SPECFUNPHYS class eulex supports the direct evaluation of Euler polynomials. The computation is not restricted to the interval [0, 1], but complex
numbers are only supported for the polynomial representation.
Methods The SPECFUNPHYS class bepolymeth comes with additional methods (plot and computing the n-th derivative) for the classes bernpoly and
eulepoly, and bexplot with an additional plot-method for bernx and
eulex.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_19
225