4
Bessel and Airy Functions
Abstract
Topics of this chapter are Bessel and Airy functions and related functions. Airy
functions and its first derivatives, its phase and modulus, and the scorer functions
H i and G i and its derivatives are discussed. In dependence of their function
argument various computational techniques will be used, e.g., series expansions
or path integration techniques of the corresponding differential equation. Similar
techniques will be also used to evaluate the Bessel functions, the modified Bessel
functions, the spherical Bessel functions, and the Kelvin functions in the complex
domain and for complex order. The corresponding programming code can be
downloaded.
In the present chapter we will discuss Airy, Bessel, and related functions. They play
an important role in physics and are used for various applications. Applications
of Airy functions in classical physics are, e.g., in optics, electromagnetism, fluid
dynamics, and in quantum dynamics and quantum optics, e.g., WKBJ approximations, systems in external homogenous electric fields, to name only a few. (Nice
visualizations can be found for Airy rainbows in the net.) Bessel functions of first
kind are widely used for solving Laplace or Helmholtz equations with cylindrical
symmetry. Bessel functions are sometimes called cylindrical functions and play also
a prominent role in describing scattering of light and electromagnetic radiation.
Spherical Bessel functions are, e.g., used in solving the Helmholtz equations in
spherical coordinates.
In the next chapter we will list the relevant functions followed by chapters about
the Airy functions and related functions and Bessel functions and related functions.
© 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_4
65
Bessel and Airy Functions
Abstract
Topics of this chapter are Bessel and Airy functions and related functions. Airy
functions and its first derivatives, its phase and modulus, and the scorer functions
H i and G i and its derivatives are discussed. In dependence of their function
argument various computational techniques will be used, e.g., series expansions
or path integration techniques of the corresponding differential equation. Similar
techniques will be also used to evaluate the Bessel functions, the modified Bessel
functions, the spherical Bessel functions, and the Kelvin functions in the complex
domain and for complex order. The corresponding programming code can be
downloaded.
In the present chapter we will discuss Airy, Bessel, and related functions. They play
an important role in physics and are used for various applications. Applications
of Airy functions in classical physics are, e.g., in optics, electromagnetism, fluid
dynamics, and in quantum dynamics and quantum optics, e.g., WKBJ approximations, systems in external homogenous electric fields, to name only a few. (Nice
visualizations can be found for Airy rainbows in the net.) Bessel functions of first
kind are widely used for solving Laplace or Helmholtz equations with cylindrical
symmetry. Bessel functions are sometimes called cylindrical functions and play also
a prominent role in describing scattering of light and electromagnetic radiation.
Spherical Bessel functions are, e.g., used in solving the Helmholtz equations in
spherical coordinates.
In the next chapter we will list the relevant functions followed by chapters about
the Airy functions and related functions and Bessel functions and related functions.
© 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_4
65