The Laplace transform: Theory and applications. (English) Zbl 0934.44001

Schiff, Joel L.

Undergraduate Texts in Mathematics. New York, NY: Springer. xiv, 233 p. (1999).

This remarkable textbook for students of either mathematics or engineering deals with a rigorous, but well readable, treatment of the basic properties of the Laplace transform and its applications to various types of equations up to partial differential equations, in particular to those stemming from physics and engineering. The necessary mathematical remedies are explained in detail, e.g. the elements of the theory of analytical functions and those of the Riemann-Stieltjes integral in order to have a rigorous basis for the treatment of the complex inversion formula and the Dirac delta function, respectively. The book contains several exercises with solutions and a table of the usual Laplace transforms.
Let us mention that \(s^{-1}\log s\) on p. 156 does satisfy (4.8), and that the complicated conditions of Theorem 4.3 are equivalent to the requirement that \(F(s)\) is a proper rational function.

Reviewer: Lothar Berg (Rostock)

MSC:

44A10	Laplace transform
44-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral transforms
35A22	Transform methods (e.g., integral transforms) applied to PDEs
00A06	Mathematics for nonmathematicians (engineering, social sciences, etc.)

Keywords:

analytic functions; transform methods; textbook; Laplace transform; partial differential equations; Riemann-Stieltjes integral; complex inversion formula; exercises