2
Error Functions and Fresnel Integrals
Abstract
Topic of this chapter is the evaluation of the error function, the Fresnel functions,
and some related functions, like the Dawson integral and the Voigt profile
functions in the complex domain. The computation is based on the incomplete
gamma function, and thus various numerical techniques are used in dependence
of the function arguments. The corresponding programming code is freely
available.
In this chapter, we will discuss the error functions, the Fresnel integrals, and some
related functions. Applications occur in a variety of physical applications, e.g., in
the solutions of diffusion problems, probability theory, the analysis of the diffraction
of light, in quest of plasma dispersion, or in spectral line profiles. First, we start with
a listed overview of relevant MATLAB and SPECFUNPHYS functions.
2.1
Function Overview
Error functions:
• erf and erfc are MATLAB functions to compute the error function and
complementary error function. Both are restricted to real values.
• The SPECFUNPHYS class erfComp allows the computation of the error and
complementary error function in the complex domain.
• The inverse of the error functions is only unique in the real domain and can
be computed with the MATLAB functions erfinv and erfcinv.
Dawson integral: The SPECFUNPHYS class dawsonC returns the Dawson integral or function.
© 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_2
23
Error Functions and Fresnel Integrals
Abstract
Topic of this chapter is the evaluation of the error function, the Fresnel functions,
and some related functions, like the Dawson integral and the Voigt profile
functions in the complex domain. The computation is based on the incomplete
gamma function, and thus various numerical techniques are used in dependence
of the function arguments. The corresponding programming code is freely
available.
In this chapter, we will discuss the error functions, the Fresnel integrals, and some
related functions. Applications occur in a variety of physical applications, e.g., in
the solutions of diffusion problems, probability theory, the analysis of the diffraction
of light, in quest of plasma dispersion, or in spectral line profiles. First, we start with
a listed overview of relevant MATLAB and SPECFUNPHYS functions.
2.1
Function Overview
Error functions:
• erf and erfc are MATLAB functions to compute the error function and
complementary error function. Both are restricted to real values.
• The SPECFUNPHYS class erfComp allows the computation of the error and
complementary error function in the complex domain.
• The inverse of the error functions is only unique in the real domain and can
be computed with the MATLAB functions erfinv and erfcinv.
Dawson integral: The SPECFUNPHYS class dawsonC returns the Dawson integral or function.
© 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_2
23