MATH2310 – DE Lecture 7
Laplace transformation:
Given a function:
f : [ 0 , ∞ ) → R (or C )
the Laplace of f (t) is the function:
∞
F ( s ) =L { f ( t ) } ( s ) =∫ e
−st
f ( t ) dt
0
provided the integral exists and is finite.
For:
n
f ( t )=t
where n ≥ 1,
n!
F ( s) = n+1
s
Laplace transformation properties:
The Laplace transformation is linear. For example, given:
L { f ( t ) } ( s )=F ( s ) , L { g ( t ) } ( s )=G( s)
then:
L { f ( t )+ g (t ) } ( s )=F ( s )+ G ( s ) ,
a L { f ( t ) } ( s )=aF ( s)
where a is any real or complex number.
Exponential Laplace transformation:
If a function f :¿ is of exponential order, i.e., there exists B , d , and T so that:
|f (t)|≤ B e dt for all t ≥ T
and the function f (t) is piecewise continuous on ¿ with only ‘finite jumps’ (no poles’,
then the Laplace transform
F ( s ) =L { f ( t ) } ( s )
Laplace transformation:
Given a function:
f : [ 0 , ∞ ) → R (or C )
the Laplace of f (t) is the function:
∞
F ( s ) =L { f ( t ) } ( s ) =∫ e
−st
f ( t ) dt
0
provided the integral exists and is finite.
For:
n
f ( t )=t
where n ≥ 1,
n!
F ( s) = n+1
s
Laplace transformation properties:
The Laplace transformation is linear. For example, given:
L { f ( t ) } ( s )=F ( s ) , L { g ( t ) } ( s )=G( s)
then:
L { f ( t )+ g (t ) } ( s )=F ( s )+ G ( s ) ,
a L { f ( t ) } ( s )=aF ( s)
where a is any real or complex number.
Exponential Laplace transformation:
If a function f :¿ is of exponential order, i.e., there exists B , d , and T so that:
|f (t)|≤ B e dt for all t ≥ T
and the function f (t) is piecewise continuous on ¿ with only ‘finite jumps’ (no poles’,
then the Laplace transform
F ( s ) =L { f ( t ) } ( s )
