Can one show that the following integral converges on $-1<\Re s < 1$ and define holomorphic function of $s$?
$$\int_0^\infty \sin(y) y^{s-1} dy$$
I've googled for a while, but I could not find any good reference.
Can one show that the following integral converges on $-1<\Re s < 1$ and define holomorphic function of $s$?
$$\int_0^\infty \sin(y) y^{s-1} dy$$
I've googled for a while, but I could not find any good reference.
by analytic continuation i think we have
$$ \int_{0}^{\infty}dtsin(t)t^{s-1} = \Gamma (s) sin( \frac{\pi s}{2})$$
the sine part comes from the imaginarhy part of $ i^{s}=e^{i\pi s} $