91Ó°ÊÓ

By considering the integral of $$ \left(\frac{\sin \alpha z}{\alpha z}\right)^{2} \frac{\pi}{\sin \pi z}, \quad \alpha<\frac{\pi}{2} $$ around a circle of large radius, prove that $$ \sum_{m=1}^{\infty}(-1)^{m-1} \frac{\sin ^{2} m \alpha}{(m \alpha)^{2}}=\frac{1}{2} $$

The Bessel function \(J_{v}(z)\) is given for \(|\arg z|<\frac{1}{2} \pi\) by the integral around a contour \(C\) of the function $$ g(z)=\frac{1}{2 \pi i} t^{-(v+1)} \exp \left[\frac{z}{2}\left(t-\frac{1}{t}\right)\right] $$ The contour starts and ends along the negative real \(t\)-axis and encircles the origin in the positive sense. It can be considered to be made up of two contours. One of them, \(C_{2}\), starts at \(t=-\infty\), runs through the third quadrant to the point \(t=-i\) and then approaches the origin in the fourth quadrant in a curve that is ultimately antiparallel to the positive real axis. The other contour, \(C_{1}\), 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. The contribution to \(J_{v}(z)\) from the curve \(C_{k}\) is \(\frac{1}{2} H_{v}^{(k)}\), the function \(H_{v}^{(k)}\) being known as a Hankel function. Using the method of steepest descents, establish the leading term in an asymptotic expansion for \(H_{v}^{(1)}\) for \(z\) real, large and positive. Deduce, without detailed calculation, the corresponding result for \(H_{v}^{(2)} .\) Hence establish the asymptotic form of \(J_{v}(z)\) for the same range of \(z\).