# limiting form as a Bessel function

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## 1—10 of 28 matching pages

##### 2: 18.34 Bessel Polynomials
18.34.8 $\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)% }{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right).$
##### 3: 10.24 Functions of Imaginary Order
###### §10.24 Functions of Imaginary Order
In consequence of (10.24.6), when $x$ is large $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either $\widetilde{J}_{\nu}\left(x\right)$ and $\tanh\left(\tfrac{1}{2}\pi\nu\right)\widetilde{Y}_{\nu}\left(x\right)$ or $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$. …
##### 4: 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 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 5: 10.45 Functions of Imaginary Order
###### §10.45 Functions of Imaginary Order
The corresponding result for $\widetilde{K}_{\nu}\left(x\right)$ is given by … 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 $\nu=0$. …
##### 6: 10.5 Wronskians and Cross-Products
###### §10.5 Wronskians and Cross-Products
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 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (1)}_{\nu+1}}\left(z\right)=2i/(\pi z),$
10.5.4 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (2)}_{\nu+1}}\left(z\right)=-2i/(\pi z),$
10.5.5 $\mathscr{W}\left\{{H^{(1)}_{\nu}}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)% \right\}={H^{(1)}_{\nu+1}}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-{H^{(1)}% _{\nu}}\left(z\right){H^{(2)}_{\nu+1}}\left(z\right)=-4i/(\pi z).$
##### 8: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
###### §33.10(ii) Large Positive $\eta$
$F_{0}\left(\eta,\rho\right)\sim e^{-\pi\eta}(\pi\rho)^{\ifrac{1}{2}}I_{1}\left% ((8\eta\rho)^{\ifrac{1}{2}}\right),$
##### 9: 10.52 Limiting Forms
###### §10.52 LimitingForms
10.52.1 $\mathsf{j}_{n}\left(z\right),{\mathsf{i}^{(1)}_{n}}\left(z\right)\sim z^{n}/(2% n+1)!!,$
10.52.2 $-\mathsf{y}_{n}\left(z\right),i{\mathsf{h}^{(1)}_{n}}\left(z\right),-i{\mathsf% {h}^{(2)}_{n}}\left(z\right),(-1)^{n}{\mathsf{i}^{(2)}_{n}}\left(z\right),(2/% \pi)\mathsf{k}_{n}\left(z\right)\sim(2n-1)!!/z^{n+1}.$
10.52.5 ${\mathsf{i}^{(1)}_{n}}\left(z\right)\sim{\mathsf{i}^{(2)}_{n}}\left(z\right)% \sim\tfrac{1}{2}z^{-1}e^{z},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$,
10.52.6 $\mathsf{k}_{n}\left(z\right)\sim\tfrac{1}{2}\pi z^{-1}e^{-z}.$
##### 10: 10.7 Limiting Forms
###### §10.7 LimitingForms
10.7.2 ${H^{(1)}_{0}}\left(z\right)\sim-{H^{(2)}_{0}}\left(z\right)\sim(2i/\pi)\ln z,$
For ${H^{(1)}_{-\nu}}\left(z\right)$ and ${H^{(2)}_{-\nu}}\left(z\right)$ when $\Re\nu>0$ combine (10.4.6) and (10.7.7). For ${H^{(1)}_{i\nu}}\left(z\right)$ and ${H^{(2)}_{i\nu}}\left(z\right)$ when $\nu\in\mathbb{R}$ and $\nu\neq 0$ combine (10.4.3), (10.7.3), and (10.7.6). … For the corresponding results for ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ see (10.2.5) and (10.2.6).