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of a function G(x). It is interesting to note that gj(t) and g,(t) are related by a formula which may be written as a Hilbert transform. Assume that -1/2<P<1/2 and that p and q are positive real numbers. Then, consider the integral fzvHvl)(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 and radius e, (iii) a small semicircle C (above the real axis) with centre the point z =p and radius i, 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 obtain'

On the Hankel 𝐽-, 𝑌- and 𝐻-transforms