This chapter deals with properties of exponential functions which are density functions for the distribution functions of important classes of observables. The integrals involved are Riemann integrals and extended Riemann integrals (in which a limiting value is obtained by letting a parameter in the domain tend to infinity), and can therefore be interpreted as Henstock integrals. The chapter uses the standard Riemann notation for the contour integrals of complex variable theory. The latter are Riemann, and therefore Riemann-complete, integrals. Variants of a theorem presented in the chapter give the indefinite integral representation of the Gaussian or Fresnel integrand, in Stieltjes or interval function form. The chapter discusses Fresnel distribution function, and investigates integrability of particular functions on R^T domain, with a view to applications in the study of random variability. The chapter derives some Fresnel continuity properties that depend on the “transition” factors or increments.

Controlled Vocabulary Terms

density function; Fresnel integrals; integration; Riemann distribution; Stieltjes integral