# of the first and second kinds

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##### 2: 10.45 Functions of Imaginary Order
10.45.3 $\displaystyle\widetilde{I}_{-\nu}\left(x\right)=\widetilde{I}_{\nu}\left(x% \right),$ $\displaystyle\widetilde{K}_{-\nu}\left(x\right)=\widetilde{K}_{\nu}\left(x% \right),$
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): … 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 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
10.34.3 $I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(\pm e^{m\nu\pi i}K_{\nu}\left(ze^% {\pm\pi i}\right)\mp e^{(m\mp 1)\nu\pi i}K_{\nu}\left(z\right)\right),$
10.34.5 $K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
##### 5: 19.4 Derivatives and Differential Equations
$\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k% }=kK\left(k\right),$
$\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k% \right)}{k},$
$\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left% (k\right)}{{k^{\prime}}^{2}},$
19.4.3 $\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},$
If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …
##### 6: 10.28 Wronskians and Cross-Products
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 7: 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 $\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}$. …
##### 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). …
##### 9: 26.1 Special Notation
 $x$ real variable. …
##### 10: 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. …