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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 0doi:10.2307/2033079 fatcat:pwv7wvthlravdlyya7lqqk3t2i