# §10.54 Integral Representations

 10.54.1 $\mathsf{j}_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos\left(z% \cos\theta\right)(\sin\theta)^{2n+1}\mathrm{d}\theta.$
 10.54.2 $\displaystyle\mathsf{j}_{n}\left(z\right)$ $\displaystyle=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P_{n}\left(\cos% \theta\right)\sin\theta\mathrm{d}\theta.$ 10.54.3 $\displaystyle\mathsf{k}_{n}\left(z\right)$ $\displaystyle=\frac{\pi}{2}\int_{1}^{\infty}e^{-zt}P_{n}\left(t\right)\mathrm{% d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi.$ 10.54.4 $\displaystyle\mathsf{j}_{n}\left(z\right)$ $\displaystyle=\frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}Q_{n}% \left(t\right)\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi.$
 10.54.5 $\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$ $\displaystyle=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}Q_{n}\left(t% \right)\mathrm{d}t,$ $\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$ $\displaystyle=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}Q_{n}\left(t% \right)\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi.$

For the Legendre polynomial $P_{n}$ and the associated Legendre function $Q_{n}$ see §§18.3 and 14.21(i), with $\mu=0$ and $\nu=n$.

Additional integral representations can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.9 and §10.32.