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21: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

22: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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23: 19.3 Graphics

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Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $K\left(k\right)$, becoming infinite when $k^2=1$. … Magnify 3D Help

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Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $E\left(k\right)$, with value 1 at $k^2=1$. … Magnify 3D Help

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. …If ${\alpha }^2=0$, then it reduces to $K\left(k\right)$. … Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …If ${\sin }^2\varphi =1$ ($>k^2$), then the function reduces to $\mathrm{\Pi }(2,k)$ with Cauchy principal value $K\left(k\right)-\mathrm{\Pi }(\frac{1}{2}{k}^2,k)$, which tends to $-\mathrm{\infty }$ as $k^2\to 1-$. …If ${\sin }^2\varphi =1/k^2$ (), then by (19.7.4) it reduces to $\mathrm{\Pi }(2/k^2,1/k)/k$, $k^2\ne 2$, with Cauchy principal value $(K\left(1/k\right)-\mathrm{\Pi }(\frac{1}{2},1/k))/k$, , by (19.6.5). … Magnify 3D Help

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24: 30.2 Differential Equations

… ►The equation contains three real parameters $\lambda$, ${\gamma }^2$, and $\mu$. … ►With $\zeta =\gamma z$ Equation (30.2.1) changes to … ►If $\gamma =0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). If ${\mu }^2=\frac{1}{4}$, Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

25: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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26: 27.16 Cryptography

… ►To code a piece $x$, raise $x$ to the power $r$ and reduce $x^r$ modulo $n$ to obtain an integer $y$ (the coded form of $x$) between $1$ and $n$. … ►In other words, to recover $x$ from $y$ we simply raise $y$ to the power $s$ and reduce modulo $n$. …

27: 14.29 Generalizations

… ►As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when ${\mu }_1={\mu }_2=\mu$. …

28: 29.11 Lamé Wave Equation

… ►In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

29: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

30: William P. Reinhardt

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20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.