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Laplace Transform Basics

Operational calculus

1. Laplace Transform Basics

Ordinary and partial differential equations describe the way certain quantities
vary with time, such as the current in an electrical circuit, the oscillations of a

vibrating membrane, or the flow of heat through an insulated conductor. These
equations are generally coupled with initial conditions that describe the state of
the system at time t = 0.
A very powerful technique for solving these problems is that of the Laplace
transform, which literally transforms the original differential equation into an

elementary algebraic expression. This latter can then simply be transformed
once again, into the solution of the original problem. This technique is known as
the “Laplace transform method.” It will be treated extensively in next sections. In
the present section we lay down the foundations of the theory and the basic

properties of the Laplace transform.
The Laplace transform is named after mathematician and astronomer Pierre-
Simon Laplace, who used a similar transform (now called z transform) in his work
on probability theory. The current widespread use of the transform came about

soonafterWorldWarIIalthoughithadbeenusedinthe19thcenturybyAbel, Lerch
and Heaviside.

Basic principles

Definition 1.1✓ Given f(t), t > 0, the Laplace transform of f(t)
which is denoted by L[f(t)] (or F(s)) is defined by

∞ A
∫ ∫
L[f(t)] = F(s) = e −st f(t) dt = lim e −st f(t) dt s > 0 (1.1)
A→∞
0 0
. . . . .

We say the transform converges if the limit exists, and diverges if not. The
notation L(f) will also be used to denote the Laplace transform of f, and the
integral is the ordinary Riemann (improper) integral.

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