# spherical (or spherical polar)

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##### 2: 10.55 Continued Fractions
###### §10.55 Continued Fractions
For continued fractions for $\mathsf{j}_{n+1}\left(z\right)/\mathsf{j}_{n}\left(z\right)$ and ${\mathsf{i}^{(1)}_{n+1}}\left(z\right)/{\mathsf{i}^{(1)}_{n}}\left(z\right)$ see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
##### 3: 10.48 Graphs
###### §10.48 Graphs Figure 10.48.5: i 0 ( 1 ) ⁡ ( x ) , i 0 ( 2 ) ⁡ ( x ) , k 0 ⁡ ( x ) , 0 ≤ x ≤ 4 . Magnify Figure 10.48.6: i 1 ( 1 ) ⁡ ( x ) , i 1 ( 2 ) ⁡ ( x ) , k 1 ⁡ ( x ) , 0 ≤ x ≤ 4 . Magnify Figure 10.48.7: i 5 ( 1 ) ⁡ ( x ) , i 5 ( 2 ) ⁡ ( x ) , k 5 ⁡ ( x ) , 0 ≤ x ≤ 8 . Magnify
##### 4: 10.50 Wronskians and Cross-Products
###### §10.50 Wronskians and Cross-Products
$\mathscr{W}\left\{{\mathsf{h}^{(1)}_{n}}\left(z\right),{\mathsf{h}^{(2)}_{n}}% \left(z\right)\right\}=-2iz^{-2}.$
$\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),{\mathsf{i}^{(2)}_{n}}% \left(z\right)\right\}=(-1)^{n+1}z^{-2},$
$\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),\mathsf{k}_{n}\left(z% \right)\right\}=\mathscr{W}\left\{{\mathsf{i}^{(2)}_{n}}\left(z\right),\mathsf% {k}_{n}\left(z\right)\right\}\\ =-\tfrac{1}{2}\pi z^{-2}.$
Results corresponding to (10.50.3) and (10.50.4) for ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and ${\mathsf{i}^{(2)}_{n}}\left(z\right)$ are obtainable via (10.47.12).
##### 7: 10.52 Limiting Forms
###### §10.52 Limiting Forms
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 $-\mathsf{y}_{n}\left(z\right),i{\mathsf{h}^{(1)}_{n}}\left(z\right),-i{\mathsf% {h}^{(2)}_{n}}\left(z\right),(-1)^{n}{\mathsf{i}^{(2)}_{n}}\left(z\right),(2/% \pi)\mathsf{k}_{n}\left(z\right)\sim(2n-1)!!/z^{n+1}.$
${\mathsf{h}^{(1)}_{n}}\left(z\right)\sim i^{-n-1}z^{-1}e^{iz},$
10.52.5 ${\mathsf{i}^{(1)}_{n}}\left(z\right)\sim{\mathsf{i}^{(2)}_{n}}\left(z\right)% \sim\tfrac{1}{2}z^{-1}e^{z},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$,
##### 8: 10.51 Recurrence Relations and Derivatives
Let $f_{n}(z)$ denote any of $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, or ${\mathsf{h}^{(2)}_{n}}\left(z\right)$. …
$nf_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_{n}^{\prime}(z),$ $n=1,2,\dots$,
$f_{n}^{\prime}(z)=-f_{n+1}(z)+(n/z)f_{n}(z),$ $n=0,1,\dots.$
Let $g_{n}(z)$ denote ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, or $(-1)^{n}$ $\mathsf{k}_{n}\left(z\right)$. Then …
##### 9: 10.57 Uniform Asymptotic Expansions for Large Order
###### §10.57 Uniform Asymptotic Expansions for Large Order
Asymptotic expansions for $\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)$, $\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, and $\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)$ as $n\to\infty$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for ${\mathsf{i}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$ the connection formula (10.47.11) is available. For the corresponding expansion for $\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)$ use
10.57.1 $\mathsf{j}_{n}'\left((n+\tfrac{1}{2})z\right)=\frac{\pi^{\frac{1}{2}}}{((2n+1)% z)^{\frac{1}{2}}}J_{n+\frac{1}{2}}'\left((n+\tfrac{1}{2})z\right)-\frac{\pi^{% \frac{1}{2}}}{((2n+1)z)^{\frac{3}{2}}}J_{n+\frac{1}{2}}\left((n+\tfrac{1}{2})z% \right).$
##### 10: 10.53 Power Series
###### §10.53 Power Series
10.53.4 ${\mathsf{i}^{(2)}_{n}}\left(z\right)=\frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}% \frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{% \infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.$
For ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ and ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For $\mathsf{k}_{n}\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).