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.

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:

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

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**Copyright **© **1996
Department
of Mathematics, Oregon State
University**

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