7. The Inverse Laplace Transform

Definition

Later, on this page...

Partial Fraction Types

Integral and Periodic Types

If G(s) = laplace{g(t)}, then the inverse transform of G(s) is defined as:

laplace-1G(s) = g(t)


Some Properties of the Inverse Laplace Transform

We first saw these properties in the Table of Laplace Transforms.


Property 1: Linearity Property

laplace-1{a G1(s) + b G2(s)} = a g1(t) + b g2(t)


Property 2: Shifting Property

If laplace-1G(s) = g(t), then laplace-1G(s - a) = eatg(t)


Property 3

If laplace-1G(s) = g(t), then MATH


Property 4

If laplace-1G(s) = g(t), then laplace-1{e-asG(s)} = u(t - a) • g(t - a)


EXAMPLES

Find the inverse of the following transforms and sketch the functions so obtained.

(a) MATH


Answer


(b) MATH


Answer


(c) $G(s)=$ MATH


Answer


(d) MATH


Answer


(e) MATH


Answer


(f) MATH


Answer


(g) MATH (where T is a constant)


Answer



Examples Involving Partial Fractions

We first met Partial Fractions in the Methods of Integration section. You may wish to revise partial fractions before attacking this section.

Obtain the inverse Laplace transforms of the following functions:

(a) MATH


Answer


(b) MATH


Answer



Integral and Periodic Types

(a) MATH


Answer


(b) MATH


Answer


(c) MATH


Answer




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