# §10.64 Integral Representations

## Schläfli-Type Integrals

 10.64.1 $\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\cos(x\sin t-nt)\cosh(x\sin t)\mathrm{d}t,$
 10.64.2 $\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\sin(x\sin t-nt)\sinh(x\sin t)\mathrm{d}t.$

See Apelblat (1991) for these results, and also for similar representations for $\operatorname{ber}_{\nu}\left(x\sqrt{2}\right)$, $\operatorname{bei}_{\nu}\left(x\sqrt{2}\right)$, and their $\nu$-derivatives.