2. A Laplace transform is a type of
integral transform.
•Plug one function in
0
s t
e dt
( )f t
•Get another function out
( )F s
•The new function is in a different domain.
3. Laplace transforms
A Laplace transform is an example of an
improper integral : one of its limits is infinite.
0 0
( ) lim ( )
h
s t s t
h
e f t dt e f t dt
Defination:
4. Dirac’s Delta Function
Mathematically
impulsive forces are
idealized by impulsive
functions which is a
discontinuous
functions whose total
value is concentrated
at one point.
The Impulse function
having magnitude 1 is
known as Dirac Delta
function or Unit
Impulse function.
7. 1
1
A sawtooth function
t
Laplace transforms are particularly effective
on differential equations with forcing functions
that are piecewise, like the Heaviside function,
and other functions that turn on and off.
X
Y
8. Top Hat function
Top Hat function is defined as follows:
◦ H(t-a)-H(t-b) =1 ; a < = t < b
=0 ; otherwise
9. Top Hat Function
Using this the Top Hat Function may
be expressed as:
F(t) = f1(t) [H(t) – H(t-t1)] + f2(t)[H(t-t1) –
H(t-t2)] + f3(t)[H(t1-t2)]
= f1(t)H(t) + [f2(t) – f1(t)]H(t-t1) +
[f3(t) – f2(t)]H(t-t2)
which is same as that of Heaviside
Step Function.
