The contribution to Jν(z) from the curve Ck is 21Hν(k), the function Hν(k) being known as a Hankel function. The contour C2 starts at t=−i∞, passes through t=−i and then approaches the origin in the fourth quadrant in a curve that is ultimately antiparallel to the positive real axis. The other contour, C1, is the mirror image of this in the real axis; it is confined to the upper half-plane, passes through t=i and is antiparallel to the real t-axis at both of its extremities.
Using the method of steepest descents, establish the leading term in an asymptotic expansion for Hν(1) for z real, large and positive. Deduce, without detailed calculation, the corresponding result for Hν(2). Hence establish the asymptotic form of Jν(z) for the same range of z.