Laplace Transforms and Their Applications to Differential Equations

N.W. McLachlan

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486798233
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This introduction to modern operational calculus offers a classic exposition of Laplace transform theory and its application to the solution of ordinary and partial differential equations. The treatment is addressed to graduate students in engineering, physics, and applied mathematics and may be used as a primary text or supplementary reading.
Chief topics include the theorems or rules of the operational calculus, evaluation of integrals and establishment of mathematical relationships, derivation of Laplace transforms of various functions, the Laplace transform for a finite interval, and other subjects. Many problems and illustrative examples appear throughout the book, which is further augmented by helpful Appendixes.

LAPLACE TRANSFORMS

AND THEIR APPLICATIONS TO DIFFERENTIAL EQUATIONS

THE LAPLACE TRANSFORM

1.11. Definition. Consider the infinite integral

p being a suitable parameter, either real or complex, while f(t) is a single-valued* function integrable in every positive interval of t. t is real and

0. This integral, but without the external p, was introduced into mathematical analysis by Laplace about the year 1779. ϕ(р), the function obtained by evaluating the integral, we define to be the p-multiplied Laplace Transform of f(t). If ϕ(p) is given, f(t) is said to be its inverse or interpretation in terms of the real variable t. When the range of integration in (1) is t = (0, + ∞), ϕ is a function of p alone. If the range is t = (h₁, h₂), 0

h₁, h₂, ϕ is a function of p, h₁ and h₂. We then write

the r.h.s. being defined as the p-multiplied L.T. of f(t) for the intervals t = (h₁, h₂). Integral (2) may be written in the form (1), for if

†

then

Integral (1) is a particular case of (2) with h₁ = 0 and h₂ → + ∞. When p is real and > 0, ‡ (1), (2) may be interpreted geometrically as the areas of the exponentially damped function p f(t) between the limits t = (0, + ∞), t = (h_l, h₂), respectively.

The p-multiplied Laplace transform of a function, as defined by (1), is usually identical with what Heaviside called its operational form,* and the latter nomenclature is used frequently. By way of variation, some writers refer to ϕ(p) as the image of the original function f(t). The most rational terminolo...