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##### 1: 10.26 Graphics Figure 10.26.1: I 0 ⁡ ( x ) , I 1 ⁡ ( x ) , K 0 ⁡ ( x ) , K 1 ⁡ ( x ) , 0 ≤ x ≤ 3 . Magnify Figure 10.26.2: e − x ⁢ I 0 ⁡ ( x ) , e − x ⁢ I 1 ⁡ ( x ) , e x ⁢ K 0 ⁡ ( x ) , e x ⁢ K 1 ⁡ ( x ) , 0 ≤ x ≤ 10 . Magnify Figure 10.26.8: I ~ 1 ⁡ ( x ) , K ~ 1 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify Figure 10.26.9: I ~ 5 ⁡ ( x ) , K ~ 5 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify Figure 10.26.10: K ~ 5 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify
##### 2: 26.21 Tables
###### §26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\genfrac{(}{)}{0.0pt}{}{m}{n}$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s\left(n,k\right)$ and $S\left(n,k\right)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p\left(\mathcal{D},n\right)$ for $n$ up to 500. … It also contains a table of Gaussian polynomials up to $\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}$. …
##### 3: 26.8 Set Partitions: Stirling Numbers
###### §26.8(i) Definitions
$S\left(n,k\right)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\ldots,n\}$ into exactly $k$ nonempty subsets. …
##### 4: 10.34 Analytic Continuation
10.34.2 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
10.34.4 $K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu% \pi\right)K_{\nu}\left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K% _{\nu}\left(z\right)\right).$
10.34.5 $K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
10.34.6 $K_{n}\left(ze^{m\pi i}\right)=\pm(-1)^{n(m-1)}mK_{n}\left(ze^{\pm\pi i}\right)% \mp(-1)^{nm}(m\mp 1)K_{n}\left(z\right).$
$K_{\nu}\left(\overline{z}\right)=\overline{K_{\nu}\left(z\right)}.$
##### 5: 10.45 Functions of Imaginary Order
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): … 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 $\nu=0$. … For graphs of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ see §10.26(iii). For properties of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). …
##### 6: 26.17 The Twelvefold Way
In this table ${\left(k\right)_{n}}$ is Pochhammer’s symbol, and $S\left(n,k\right)$ and $p_{k}\left(n\right)$ are defined in §§26.8(i) and 26.9(i). …
##### 7: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 8: 19.4 Derivatives and Differential Equations
$\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k% \right)}{k},$
$\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left% (k\right)}{{k^{\prime}}^{2}},$
19.4.3 $\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},$
19.4.6 $\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F% \left(\phi,k\right)}{k},$
If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …
##### 9: 30.5 Functions of the Second Kind
###### §30.5 Functions of the Second Kind
30.5.1 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$ $n=m,m+1,m+2,\dots$.
30.5.2 $\mathsf{Qs}^{m}_{n}\left(-x,\gamma^{2}\right)=(-1)^{n-m+1}\mathsf{Qs}^{m}_{n}% \left(x,\gamma^{2}\right),$
30.5.3 $\mathsf{Qs}^{m}_{n}\left(x,0\right)=\mathsf{Q}^{m}_{n}\left(x\right);$
30.5.4 $\mathscr{W}\left\{\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),\mathsf{Qs}^{m}% _{n}\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})\quad(\neq 0),$
##### 10: 28.5 Second Solutions $\operatorname{fe}_{n}$, $\operatorname{ge}_{n}$
Second solutions of (28.2.1) are given by … …