What is the value of the Mellin transform ?

[zetafunction]

my question is what is the value of

\int_{0}^{\infty}dtt^{s-1}

if s >1 the integral is divergent as x -->oo

if s<1 the integral would become divergent as x-->0

then what is the value of this integral, or the value (regularized i presume) of


\int_{0}^{\infty}dtt^{s-1}cos(at) fro every real value of 'a' thanks.

 

[mathman]

It has meaning only for 0<s<1. For s = 1/2, the value is √(π/2a). Since cos is even, a >0 can be assumed.

 

[zetafunction]

thanks.. but could you use analytic continuation for values bigger than Re (s) >1 ??

for example if we consider in a regularized sense that \int_{0}^{\infty}t^{s-1}dt=0 for every 's' then

\int_{0}^{\infty}frac(1/t)t^{s-1}dt= -\zeta(s)/s for every s except s=1

 

[jostpuur]

my question is what is the value of

\int_{0}^{\infty}dtt^{s-1}

if s >1 the integral is divergent as x -->oo

if s<1 the integral would become divergent as x-->0

It is divergent on the limit x\to\infty when s\geq 0, and on the limit x\to 0^+ when s\leq 0. The whole integral is thus divergent for any s.

thanks.. but could you use analytic continuation for values bigger than Re (s) >1 ??

for example if we consider in a regularized sense that \int_{0}^{\infty}t^{s-1}dt=0 for every 's' then

What do you mean by this "regularization". You have given value 0 to a divergent integral!