$J_n(z)=(z/2)^n\frac{1}{\pi(\frac{1}{2})_n}\int_{0}^{\pi}\cos(z\cos\theta)\sin^{2n}\theta d\theta$

Question

In "A treatise on the theory of Bessel functions, Watson, p.48" he ends up with this relation for the Bessel function: $$J_n(z)=\frac{(1/2\cdot z)^n}{\Gamma(n+1/2)\Gamma(1/2)}\int_{0}^{1}t^{n-1/2}\left [ \sum_{m=0}^{\infty}\frac{(-1)^mz^{2m}(1-t)^{m-1/2}}{2m!} \right ]dt.$$ How can Ι use this to get this identity for Bessel functions? $$J_n(z)=(z/2)^n\frac{1}{\pi(\frac{1}{2})_n}\int_{0}^{\pi}\cos(z\cos\theta)\sin^{2n}\theta d\theta$$

Answer 1

Notice that \begin{align*} \sum\limits_{m = 0}^\infty {\frac{{( - 1)^m z^{2m} (1 - t)^{m - 1/2} }}{{(2m)!}}} & = (1 - t)^{ - 1/2} \sum\limits_{m = 0}^\infty {\frac{{( - 1)^m (z\sqrt {1 - t} )^{2m} }}{{(2m)!}}} \\ & = (1 - t)^{ - 1/2} \cos (z\sqrt {1 - t} ) \end{align*} and perform the substitution $t=\sin^2 \theta$, $0<\theta<\frac{\pi}{2}$. This, together with the definition of the Pochhammer symbol, gives $$ J_n (z) = \left( {\frac{z}{2}} \right)^n \frac{1}{{\pi \left( {\frac{1}{2}} \right)_n }}2\int_0^{\pi /2} {\cos (z\cos \theta )\sin ^{2n} \theta d\theta } . $$ Finally, \begin{align*} \int_0^{\pi /2} {\cos (z\cos \theta )\sin ^{2n} \theta d\theta } & = \int_0^{\pi /2} {\cos (z\cos (\pi - \theta ))\sin ^{2n} (\pi - \theta )d\theta } \\ & = \int_{\pi /2}^\pi {\cos (z\cos \varphi )\sin ^{2n} \varphi d\varphi } \end{align*} shows that $$ 2\int_0^{\pi /2} {\cos (z\cos \theta )\sin ^{2n} \theta d\theta } = \int_0^\pi {\cos (z\cos \theta )\sin ^{2n} \theta d\theta } . $$