Operational Calculus in Two Variables and Its Applications

By V.A. Ditkin and A.P. Prudnikov

A concise monograph by two Russian experts provides an account of the operational calculus in two variables based on the two-dimensional Laplace transform. Suitable for advanced undergraduates and graduate students in mathematics, the treatment requires some familiarity with operational calculus in one variable.
Part One of the two-part approach presents the fundamental theory in two chapters, examining the two-dimensional Laplace transform and offering basic definitions and theorems of the operational calculus in two variables and its applications. Part Two presents tables of formulae for various categories of functions, including rational and irrational functions; exponential and logarithmic functions; cylinder, integral, and confluent hypergeometric functions; and other areas.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 15, 2017
• ISBN: 9780486823416

Book preview

Translator.

PART ONE

Fundamental Theory

CHAPTER 1

THE TWO-DIMENSIONAL LAPLACE TRANSFORM

The well-known integral transform of Laplace was apparently considered for the first time in 1812 [40], Since then many papers have been devoted to the study of the properties of this transform and its numerous applications, and the idea naturally arose of generalizing the transform to functions of two variables. During the 1930s short notes [33, 34, 35] on the operational calculus in two variables based on the two-dimensional Laplace transform appeared. In the works of Delerue [19] and Doetsch [22] the methods of the operational calculus in several variables were successfully applied to the solution of differential equations by the study of the properties of special functions etc. Indian mathematicians have contributed several papers on this same theme [10, 11, 55]. In recent times Soviet physicists have applied the operational methods of A.V.Lykov to the analytic investigation of processes of heat and mass transfer [4, 5].

We discuss the fundamental properties of the two- dimensional Laplace transform as the basis of an operational calculus in two variables.

1. The Laplace Integral

We suppose that f(x, y) is a real or complex valued function of two real variables, defined on the region R x y < < ∞) and integrable in the sense of Lebesgue over an arbitrary finite rectangle Ra, bx ay b)).

We shall consider the expression

where p =σ + μ and q = τ + iν are complex parameters determining a point (p; q) in the plane of two complex dimensions. Let S be the class of all functions f(x, y), such that the following conditions are satisfied for at least one point(p, q):

1. The integral (1.1) is bounded at the point (p, q) with respect to the variables a and b; i.e.,

for all a 0, b 0, where M (p, q) is a positive constant independent of a and b.

2. At the point (p, q)

exists. We denote this limit by

The integral (1.2) is called the two-dimensional Laplace transform (or integral) of the function f(x, y).

If the conditions 1 and 2 are satisfied simultaneously, we will say that the integral (1.2) converges boundedly in at least one point (p, q) Thus the class S consists of functions for which the integral (1.2) converges boundedly for at least one point (p, q). When the integral (1.2) converges boundedly, we will call f(x, y) the determining function and F(p, q) the generating function.¹

Remark 1. If the function f(x, y) satisfies the condition

for all x 0, y 0 (where M, h, k are positive constants), then it is not difficult to verify that f (x, y) belongs to the class S at all points (p, q) for which Re p > h, Re q > k.

Remark 2. If the function f(x, y) — f1 (x) f2 (y) and the integrals

exist then f(x, y) belongs to the class S and F(p, q) = F1 (p) F2(q).

Theorem 1. If the integral (1.2) converges boundedly at the point (p0, q0), then it converges boundedly at all points (p, q) for which Re(p — p0)>0, Re(q — q0) > 0.

Proof. Integration by parts we obtain:

Writing

this becomes