The Laplace Transform of a Function

The Laplace Transform of a function y(t) is defined by

if the integral exists. The notation L[y(t)](s) means take the Laplace transform
of y(t).  The functions y(t) and Y(s) are partner functions. Note that Y(s) is indeed only a function of s since the definite integral is with respect to t.

Examples

Let y(t)=exp(t). We have

The integral converges if s>1. The functions exp(t) and 1/(s-1) are partner
functions.

Let y(t)=cos(3t). We have

The integral converges for s>0. The integral can be computed by doing
integration by parts twice or by looking in an integration table.

Existence of the Laplace Transform

If y(t) is piecewise continuous for t>=0 and of exponential order, then
the Laplace Transform exists for some values of s. A function y(t) is of
exponential order c if there is exist constants M and T such that

All polynomials, simple exponentials (exp(at), where a is a constant), sine
and cosine functions, and products of these functions are of exponential order.
An example of a function not of exponential order is exp(t^2). This function
grows too rapidly. The integral

does not converge for any value of s.

Table of Laplace Transforms

The following table lists the Laplace Transforms for a selection of functions

Rules for Computing Laplace Transforms of Functions

There are several formulas and properties of the Laplace transform which can
greatly simplify calculation of the Laplace transform of functions. We
summarize them below. The properties can verified using the integral
formula for the Laplace Transform and can be found in any textbook.

Linearity

Like differentiation and integration the Laplace transformation is a linear
operation. What does this mean? In words, it means that the Laplace
transform of a constant times a function is the constant times the
Laplace transform of the function. In addition the Laplace transform
of a sum of functions is the sum of the Laplace transforms.

Let us restate the above in mathspeak. Let Y_1(s) and Y_2(s) denote
the Laplace transforms of y_1(t) and y_2(t), respectively, and let c_1
be a constant. Recall that L[f(t)](s) denotes the Laplace transform of
f(t). We have

As a corollary, we have the third formula:

Here are several examples:

Here we have used the results in the table for the Laplace transform
of the exponential. Here are a couple of more examples:

Translation Property

The translation formula states that Y(s) is the Laplace transform of y(t), then

where a is a constant. Here is an example. The Laplace transform of the
y(t)=t is Y(s)=1/s^2. Hence

Laplace Transform of the Derivative

Suppose that the Laplace transform of y(t) is Y(s). Then the Laplace
Transform of y'(t) is

For the second derivative we have

For the n'th derivative we have

Derivatives of the Laplace Transform

Let Y(s) be the Laplace Transform of y(t). Then

Here is an example. Suppose we wish to compute the Laplace
transform of tsin(t). The Laplace transform of sin(t) is 1/(s^2+1).
Hence, we have

Go on to the Inverse Laplace Transform

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