A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Chapter 6

Gaussian Integrals

This chapter deals with properties of exponential functions which are density functions for the distribution functions of important classes of observables. (Density function is a point function for which the distribution function is the Riemann-complete indefinite integral.)

6.1   Fresnel’s Integral

The integrals

are well known in their familiar notation,

The latter integrals exist as improper or extended Riemann integrals. The familiar proof of the first one runs as follows.

Lemma 10

Proof. Let ρ > 0 and let

By Theorem 46 (integration by substitution), the Riemann integral

can be evaluated using polar co-ordinates (r, θ):

Letting ρ → ∞,

For λ = 1,2,3,… let J^λ = [−λ, λ] × [−λ, λ]. Then

If equation (6.2) is correct, it implies that β_λ converges as λ → ∞, ...