Change of Scale Property | Laplace Transform

Change of Scale Property
If   L{f(t)}=F(s),   then,
 

L{f(at)}=1aF(sa)

 

Proof of Change of Scale Property
L{f(at)}=0estf(at)dt
 

Let
z=at

t=z/a

dt=dza
 

when   t=0,z=0

when   t=,z=
 

L{f(at)}=0es(z/a)f(z)dza

L{f(at)}=1a0e(s/a)zf(z)dz
 

Hence,
L{f(at)}=1aF(sa)       okay