On the Hankel J-, Y- and H-Transforms

James L. Griffith
1958 Proceedings of the American Mathematical Society  
Consider the Hankel transforms /> CO xJ,(tx)G(x)dx 0 and (2) of a function G(x). It is interesting to note that gj(%) and gy(t;) are related by a formula which may be written as a Hilbert transform. Assume that -1/2 0< 1/2 and that p and q are positive real numbers. Then, consider the integral $z"H'?)(qz)(p -z)~1dz taken around a contour consisting of (i) a large semicircle (above the real axis) with centre the origin 0 and radius R, (ii) a small semicircle (above the real axis) with centre 0
more » ... is) with centre 0 and radius e, (iii) a small semicircle C (above the real axis) with centre the point z=p and radius t], and (iv) the parts of the real axis joining the ends of these semicircles. Using well known properties of the Bessel functions concerned, and in particular Watson [2, p. 75(5)] we easily obtain1
doi:10.2307/2033079 fatcat:pwv7wvthlravdlyya7lqqk3t2i