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)}=∫∞0e−stf(at)dt
Let
z=at
t=z/a
dt=dza
when t=0,z=0
when t=∞,z=∞
L{f(at)}=∫∞0e−s(z/a)f(z)dza
L{f(at)}=1a∫∞0e−(s/a)zf(z)dz
Hence,
L{f(at)}=1aF(sa) okay