Can someone give me some help on this problem?
Show that \mu(cos)(z)=\int_0^{\infty}cos(t)t^{z-1}dt=\Gamma(z)cos(\pi z/2) for 0<Re(z)<1.
By definition, \mu(f)(z)=F(z)=\int_0^{\infty}f(t)t^{z-1}dt
If I know that \mu(sin)(z)=\int_0^{\infty}sin(t)t^{z-1}dt=\Gamma(z)sin(\pi z/2) for 0<Re(z)<1, how can I show that this identity still holds in the larger strip -1<Re(z)<1