# limiting forms as trigonometric functions

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##### 2: 22.5 Special Values
###### §22.5 Special Values
For the other nine functions ratios can be taken; compare (22.2.10). …
###### §22.5(ii) Limiting Values of $k$
In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
##### 3: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
###### §33.10(i) Large $\rho$
$F_{\ell}\left(\eta,\rho\right)=\sin\left({\theta_{\ell}}\left(\eta,\rho\right)% \right)+o\left(1\right),$
##### 4: 10.45 Functions of Imaginary Order
###### §10.45 Functions of Imaginary Order
In consequence of (10.45.5)–(10.45.7), $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either $\widetilde{I}_{\nu}\left(x\right)$ and $(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)$, or $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu\neq 0$ or $\nu=0$. … In this reference $\widetilde{I}_{\nu}\left(x\right)$ is denoted by $(1/\pi)\sinh\left(\pi\nu\right)L_{i\nu}(x)$. …
##### 5: 10.24 Functions of Imaginary Order
$\widetilde{Y}_{\nu}\left(x\right)=\sqrt{2/(\pi x)}\sin\left(x-\tfrac{1}{4}\pi% \right)+O\left(x^{-\frac{3}{2}}\right).$
##### 6: 10.28 Wronskians and Cross-Products
###### §10.28 Wronskians and Cross-Products
10.28.1 $\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 7: 11.9 Lommel Functions
###### §11.9 Lommel Functions
The inhomogeneous Bessel differential equation … the right-hand side being replaced by its limiting form when $\mu\pm\nu$ is an odd negative integer. …
##### 8: 10.5 Wronskians and Cross-Products
###### §10.5 Wronskians and Cross-Products
10.5.1 $\mathscr{W}\left\{J_{\nu}\left(z\right),J_{-\nu}\left(z\right)\right\}=J_{\nu+% 1}\left(z\right)J_{-\nu}\left(z\right)+J_{\nu}\left(z\right)J_{-\nu-1}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.5.2 $\mathscr{W}\left\{J_{\nu}\left(z\right),Y_{\nu}\left(z\right)\right\}=J_{\nu+1% }\left(z\right)Y_{\nu}\left(z\right)-J_{\nu}\left(z\right)Y_{\nu+1}\left(z% \right)=2/(\pi z),$
10.5.3 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (1)}_{\nu+1}}\left(z\right)=2i/(\pi z),$
10.5.5 $\mathscr{W}\left\{{H^{(1)}_{\nu}}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)% \right\}={H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-{H^{(1)}% _{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-4i/(\pi z).$
##### 9: 6.2 Definitions and Interrelations
This is also true of the functions $\mathrm{Ci}\left(z\right)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii). … The logarithmic integral is defined by … $\mathrm{Si}\left(z\right)$ is an odd entire function. …$\mathrm{Cin}\left(z\right)$ is an even entire function. …