Question

state second shifting property​

Answer 1

state second shifting property

Second Shifting Property

If L{f(t)}=F(s),

and

g(t)={f(t−a)0t>at<a

then,

L{g(t)}=e−asF(s)

_______________________

Explaination

Proof of Second Shifting Property

g(t)={f(t−a)0t>at<a

L{g(t)}=∫∞0e−stg(t)dt

L{g(t)}=∫a0e−st(0)dt+∫∞ae−stf(t−a)dt

L{g(t)}=∫∞ae−stf(t−a)dt

Let

z=t−a

t=z+a

dt=dz

when t=a,z=0

when t=∞,z=∞

L{g(t)}=∫∞0e−s(z+a)f(z)dz

L{g(t)}=∫∞0e−sz−saf(z)dz

L{g(t)}=∫∞0e−sze−saf(z)dz

L{g(t)}=e−sa∫∞0e−szf(z)dz

L{g(t)}=e−asL{f(z)}

L{g(t)}=e−asL{f(t−a)}

L{g(t)}=e−asF(s)

___________________________

Hope it helps❤