# hyperbolic analog

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## 5 matching pages

##### 3: 18.17 Integrals
For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). …
18.17.8 $\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}{\mathrm{e}}^{x^{2}}\left(V% \left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac{2^{n+\frac{3}{2% }}n!\,{\mathrm{e}}^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{{\mathrm{e}}^{-(2n+1)t+% x^{2}\tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\,\mathrm{d}t.$
##### 4: 10.20 Uniform Asymptotic Expansions for Large Order
Note: Another way of arranging the above formulas for the coefficients $A_{k}(\zeta),B_{k}(\zeta),C_{k}(\zeta)$, and $D_{k}(\zeta)$ would be by analogy with (12.10.42) and (12.10.46). …
10.20.17 $z=\pm(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}\pm\mathrm{i}(\tau^{2}-\tau\tanh% \tau)^{\frac{1}{2}},$ $0\leq\tau\leq\tau_{0}$,
where $\tau_{0}=1.19968\ldots$ is the positive root of the equation $\tau=\coth\tau$. …
##### 5: 19.25 Relations to Other Functions
19.25.16 $\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}\omega^{2}R_{J}\left(c-1,c-k^{2% },c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{2})}{(\alpha^{2}-1)(1-\omega^{2% })}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),(\alpha^{2}-c)(c-\omega^{2})% \right),$ $\omega^{2}=k^{2}/\alpha^{2}$.
19.25.30 $\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),$
For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9). …