# §10.54 Integral Representations

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

For the Legendre polynomial $\mathop{P_{n}\/}\nolimits$ and the associated Legendre function $\mathop{Q_{n}\/}\nolimits$ 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.