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
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.
Let y(t)=exp(t). We have
The integral converges if s>1. The functions exp(t) and
1/(s-1) are partner
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
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
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
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
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
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
transform of tsin(t). The Laplace transform of sin(t) is 1/(s^2+1).
Hence, we have
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