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1: 7.23 Tables

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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2: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. … ►where $u_s(t)$ and $v_s(t)$ are polynomials in $t$ of degree $3s$, ($s$ odd), $3s-2$ ($s$ even, $s\ge 2$). … ►and the coefficients ${𝖠}_s(\tau )$ are the product of ${\tau }^s$ and a polynomial in $\tau$ of degree $2s$. … ►The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu \to \mathrm{\infty }$ uniformly with respect to $t\in [1+\delta ,\mathrm{\infty })$. … ►where $\xi ,\eta$ are given by (12.10.7), (12.10.23), respectively, and …

3: William P. Reinhardt

… ►His undergraduate and graduate degrees are from the University of California at Berkeley and Harvard University, respectively. …Reinhardt is a frequent visitor to the NIST Physics Laboratory in Gaithersburg, and to the Joint Quantum Institute (JQI) and Institute for Physical Sciences and Technology (ISTP) at the University of Maryland. … ►He has recently carried out research on non-linear dynamics of Bose–Einstein condensates that served to motivate his interest in elliptic functions. … ►

20 Theta Functions

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

4: 28.35 Tables

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Ince (1932) includes eigenvalues $a_n$, $b_n$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also ${\mathrm{ce}}_n(x,q)$, ${\mathrm{se}}_n(x,q)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $a_n={\mathrm{𝑏𝑒}}_n-2q$, $b_n={\mathrm{𝑏𝑜}}_n-2q$.

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Kirkpatrick (1960) contains tables of the modified functions ${\mathrm{Ce}}_n(x,q)$, ${\mathrm{Se}}_{n+1}(x,q)$ for $n=0(1)5$, $q=1(1)20$, $x=0.1(.1)1$; 4D or 5D.

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National Bureau of Standards (1967) includes the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for ${\mathrm{ce}}_n(x,q)$ and ${\mathrm{se}}_n(x,q)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{e,n}(\sqrt{q})$, $f_{e,n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\text{ <span class="ltx\_text hilite">to</span> }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_n\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_n(x,10)$ for $n=1(1)10$, and the first 5 zeros of ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_n^{(j)}(x,\sqrt{10})$ for $n=0$ or $1(1)8$, $j=1,2$. Precision is mostly 9S.

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5: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

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Chazy’s second-degree Painlevé equations. J. Phys. A 39 (39), pp. 11955–11971.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§32.7(vii).

Encodings:

BibTeX

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6: 32.8 Rational Solutions

… ►Next, let $p_m(z)$ be the polynomials defined by $p_m(z)=0$ for , and … ►where $P_m(z)$ and $Q_m(z)$ are polynomials of degree $m$, with no common zeros. … ►where $P_{j,n-1}(z)$ and $Q_{j,n}(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. … ►where $\lambda$, $\mu$ are constants, and $P_{n-1}(z)$, $Q_n(z)$ are polynomials of degrees $n-1$ and $n$, respectively, with no common zeros. … ►For the case $\delta =0$ see Airault (1979) and Lukaševič (1968). …

7: 19.36 Methods of Computation

… ►For both $R_D$ and $R_J$ the factor ${(r/4)}^{-1/6}$ in Carlson (1995, (2.18)) is changed to ${(r/5)}^{-1/8}$ when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: …Polynomials of still higher degree can be obtained from (19.19.5) and (19.19.7). … ►As $n\to \mathrm{\infty }$, $c_n$, $a_n$, and $t_n$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence … ►(19.22.20) reduces to $0=0$ if $p=x$ or $p=y$, and (19.22.19) reduces to $0=0$ if $z=x$ or $z=y$. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

8: 18.39 Applications in the Physical Sciences

… ►which in one dimensional systems are typically non-degenerate, namely there is only a single eigenfunction corresponding to each ${ϵ}_n$, $n\ge 0$. …${ϵ}_0$ is referred to as the ground state, all others, $n=1,2,\mathrm{\dots },$ in order of increasing energy being excited states. … ►Namely the $k$th eigenfunction, listed in order of increasing eigenvalues, starting at $k=0$, has precisely $k$ nodes, as real zeros of wave-functions away from boundaries are often referred to. … ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure $r^{2}dr\sin \theta d\theta d\varphi$. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …