The Laplace Transform64.2 Definition of a Laplace transformThe Laplace transform of the function f (t) is definedby the integral where in general s is a complex variable.Common notations used for the Laplace transformThere are various commonly used notations for theLaplace transform of f (t) and these include:

Also, the letter p is sometimes used instead of s asthe parameter. The notation adopted in this book willbe f (t) for the original function and L{ f (t)} for itsLaplace transform.Hence, from above:

64.3 Linearity property of theLaplace transformFrom equation (1),

i.e L{k f (t)}=kL{ f (t)} (2)where k is any constant.

where a and b are any real constants.The Laplace transform is termed a linear operatorbecause of the properties shown in equations (2)and (3).64.4 Laplace transforms of elementaryFunctions(a) f (t)=1. From equation (1),

A list of elementary standard Laplace transforms aresummarized in Table 64.1.

64.5 Worked problems on standardLaplace transformsProblem 1. Using a standard list ofLaplace transforms determine the following:

Problem 2. Find the Laplace transforms of:(a) 6 sin 3t 4 cos 5t (b) 2 cosh 2 sinh 3.

Problem 4. Determine the Laplace transformsof: (a) sin2 t (b) cosh2 3x.

Problem 5. Find the Laplace transform of3 sin (t +), where and are constants.Using the compound angle formula for sin(A+B),from Chapter 18, sin(t +) may be expanded to

(sin t cos +cos t sin ). Hence

Determine the Laplace transforms in Problems1 to 9.

Properties of Laplace transforms65.1 The Laplace transform of eat f (t)From Chapter 64, the definition of the Laplacetransform of f (t) is:

Hence the substitution of (sa) for s in the transformshown in equation (1) corresponds to themultiplication of the original function f (t) by eat .This is known as a shift theorem.

65.3 The Laplace transforms ofderivatives(a) First derivativeLet the first derivative of f (t) be f _(t) then, fromequation (1),

(b) Second derivative

65.4 The initial and final valueTheorems

Scaling theorem

The convolution theorem

Inverse Laplace transformsDefinition of the inverse LaplacetransformIf the Laplace transform of a function f (t) is F(s),i.e. L{ f (t)}=F(s), then f (t) is called the inverseLaplace transform of F(s) and is written asf (t)= L1{F(s)}.

66.3 Inverse Laplace transforms usingpartial fractionsSometimes the function whose inverse is requiredis not recognisable as a standard type, such as thoselisted in Table 66.1. In such cases it may be possible,by using partial fractions, to resolve the function intosimpler fractions which may be inverted on sight.For example, the function,

cannot be inverted on sight from Table 66.1. However,by using partial fractions,

which may be inverted as1+e3t from (i) and (iii) of Table 64.1.