The integral transform
$$C_\nu(z)=\cos p\pi J_\nu(z)+\sin p\pi Y_\nu(z),$$
and $J_\nu(z)$ and $Y_\nu(z)$ are the Bessel functions of the first and second kinds, respectively. For $p=0$ the Hardy transform coincides with one of the forms of the Hankel transform, and for $p=1/2$ with the $Y$-transform. The Hardy transform was proposed by G.H. Hardy in .
The inversion formula is
The Hardy transform is also defined for certain classes of generalized functions.
|||G.H. Hardy, "Some formulae in the theory of Bessel functions" Proc. London. Math. Soc. (2) , 23 (1925) pp. 61–63|
|||Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)|
Hardy transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_transform&oldid=32672