# of the second kind

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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 10.26.7: I ~ 1 / 2 ⁡ ( x ) , K ~ 1 / 2 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . 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: 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). …
##### 3: 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 $K_{n}\left(ze^{m\pi i}\right)=\pm(-1)^{n(m-1)}mK_{n}\left(ze^{\pm\pi i}\right)% \mp(-1)^{nm}(m\mp 1)K_{n}\left(z\right).$
$K_{\nu}\left(\overline{z}\right)=\overline{K_{\nu}\left(z\right)}.$
##### 4: 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). …
##### 5: 26.8 Set Partitions: Stirling Numbers
$S\left(n,k\right)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\ldots,n\}$ into exactly $k$ nonempty subsets. …
$S\left(n,0\right)=0,$
$S\left(n,1\right)=1,$
26.8.22 $S\left(n,k\right)=kS\left(n-1,k\right)+S\left(n-1,k-1\right),$
##### 6: 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 $E\left(k\right)$. …
##### 7: 10.37 Inequalities; Monotonicity
If $\nu$ $(\geq 0)$ is fixed, then throughout the interval $0, $I_{\nu}\left(x\right)$ is positive and increasing, and $K_{\nu}\left(x\right)$ is positive and decreasing. If $x$ $(>0)$ is fixed, then throughout the interval $0<\nu<\infty$, $I_{\nu}\left(x\right)$ is decreasing, and $K_{\nu}\left(x\right)$ is increasing. …
##### 8: 10.42 Zeros
Properties of the zeros of $I_{\nu}\left(z\right)$ and $K_{\nu}\left(z\right)$ may be deduced from those of $J_{\nu}\left(z\right)$ and ${H^{(1)}_{\nu}}\left(z\right)$, respectively, by application of the transformations (10.27.6) and (10.27.8). … The distribution of the zeros of $K_{n}\left(nz\right)$ in the sector $-\tfrac{3}{2}\pi\leq\operatorname{ph}z\leq\tfrac{1}{2}\pi$ in the cases $n=1,5,10$ is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle $-\tfrac{1}{2}\pi$ so that in each case the cut lies along the positive imaginary axis. … $K_{n}\left(z\right)$ has no zeros in the sector $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$; this result remains true when $n$ is replaced by any real number $\nu$. For the number of zeros of $K_{\nu}\left(z\right)$ in the sector $|\operatorname{ph}z|\leq\pi$, when $\nu$ is real, see Watson (1944, pp. 511–513). For $z$-zeros of $K_{\nu}\left(z\right)$, with complex $\nu$, see Ferreira and Sesma (2008). …
##### 9: 10.28 Wronskians and Cross-Products
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 10: 19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^{2}$ with $0\leq m<1$ can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for $0 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …