Created by T. Madas
HEAVISIDE FUNCTION
Created by T. Madas
Created by T. Madas
Question 1 The Heaviside function H(t) is defined as
1t ≥ 0 H()t = 0t < 0
Determine the Laplace transform of H(t− c ) .
e−cs L()H()t− c = s
Created by T. Madas
Created by T. Madas
Question 2 The Heaviside step function H(t) is defined as
1t ≥ 0 H()t = 0t < 0
Determine the Laplace transform of H(t− cft) ( − c ) , where f( t ) is a continuous or piecewise continuous function defined for t ≥ 0 .
L()H()()tcftc− − = e −cs L () ft()
Created by T. Madas
Created by T. Madas
Question 3 The piecewise continuous function f( t ) is defined as
4 0≤t ≤ 2 f() t = 124−t 2 < t ≤ 4 t−8 t > 4
a) Sketch the graph of f( t ) .
b) Express f( t ) in terms of the Heaviside step function, and hence find the Laplace transform of f( t ) .
MM3-B , 8 4e−2s 5e − 4 s ft()()()()()()=4H tt −− 4 2H t −+− 25 t 4H t − 4 , L()f() t = − + s s2 s 2
Created by T. Madas
Created by T. Madas
Question 4 The piecewise continuous function f( t ) is defined as
8 0≤t ≤ 4 12−t 4 < t ≤ 6 = f() t 6 6
Express f( t ) in terms of the Heaviside step function, and hence find the Laplace transform of f( t ) .
ft=8H tt −− 4H t −+− 4 t 4H t −−− 61 t 10H t − 10 , ()()()()()() 2 ()() 8e−4s e − 6 s e − 10 s L()f() t =− + − s s2 s 22 s 2
Created by T. Madas
Created by T. Madas
Question 5 The piecewise continuous function f( t ) is defined as
72−t 0 < t ≤ 3 1 3
Express f( t ) in terms of the Heaviside step function, and hence find the Laplace transform of f( t ) .
ft()()()()()()()()()()=−72H ttttttt +− 2 3H −+− 3 7H −−− 7 15H t −− 159H t − 15 , s(79e−−15s) −+ 22e −−− 3 sss +− e 7 e 15 L()f() t = s2
Created by T. Madas
Created by T. Madas
Question 6 The piecewise continuous function f( t ) is defined as
2 3e−s 3e − 3 s f t =L−1 + − . () 2 2 s s s
Sketch the graph of f( t ) .
graph
Created by T. Madas
Created by T. Madas
Question 7 The piecewise continuous function f( t ) is defined as
e−2s f t=L−13e s 2 s − 4e s + 5 . () 2 () s
a) Determine an expression for f( t ) .
b) Sketch the graph of f( t ) .
ft()()()()()()=3H tt −− 4 1H t −+− 15 t 2H t − 2
Created by T. Madas
Created by T. Madas
Question 8 The piecewise continuous function f( t ) is defined as
−11 − 4s − 6 s f() t =L (2 − 2e − e ) . s2
Sketch the graph of f( t ).
MM3-A , graph
Created by T. Madas
Created by T. Madas
Question 9 The piecewise continuous function f( t ) is defined as
2− 3e−4s + e − 8 s f t = L−1 . () 2 s
Sketch the graph of f( t ) .
graph
Created by T. Madas
Created by T. Madas
Question 10 The piecewise continuous function f( t ) is defined as
s(79e−−15s) −+ 22e −−− 3 sss +− e 7 e 15 f() t = L−1 . s2
Sketch the graph of f( t ) .
graph
Created by T. Madas
Created by T. Madas
Question 11 The piecewise continuous function f( t ) is defined as
(1e−−2s)( 1e + − 4 s ) f() t = L−1 . s2
Sketch the graph of f( t ) .
graph
Created by T. Madas
Created by T. Madas
Question 12 The Heaviside function H( x) is defined by
1x ≥ 0 H(x ) = 0x < 0
By introducing the converging factor e−ε x and letting ε → 0 , determine the Fourier transform of H( x) .
1 ε You may assume that δ ()t = lim . π ε →0 ε 2+ t 2
1 i F []H(x ) =πδ () k − 2π k
Created by T. Madas
Created by T. Madas
Question 13 The function f is defined by
1x > 0 f() x=sign( x ) = −1x < 0
a) By introducing the converging factor e−ε x and letting ε → 0 , find the Fourier transform of f .
b) By introducing the converging factor e−ε x and letting ε → 0 , find the Fourier transform of g( x ) =1.
1 ε You may assume that δ ()t = lim . π ε →0 ε 2+ t 2
c) Hence determine the Fourier transform of the Heaviside function H( x) ,
1x ≥ 0 H(x ) = 0x < 0
i 1 1 i F []sign(x ) = − , F []1= 2 π δ ()k , F []H(x ) =πδ () k − k π 2π k
Created by T. Madas