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21: 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 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 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►They are valid over parts of the complex $k$ and $\varphi$ planes. …

22: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

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On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.

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ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt

Referenced by:

§25.10(ii), §25.18(ii).

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Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Associated computer program available online

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ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

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MathReview (J. V. Wehausen), ZentralBlatt

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2nd item.

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Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A 57 (1), pp. 190–197.

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FullText

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§5.20.

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23: Bibliography M

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Hypergeometric Functions.

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Unpublished Lecture Notes, University of London. An online version exists.

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FullText

Referenced by:

§35.7(ii), §35.8(iii).

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On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

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MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§16.21.

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

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On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

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24: 35.2 Laplace Transform

… ►where the integration variable $𝐗$ ranges over the space $𝛀$. … ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►Assume that ${\int }_{𝓢}\left|g(𝐔+\mathrm{i}𝐕)\right|d𝐕$ converges, and also that its limit as $𝐔\to \mathrm{\infty }$ is $0$. …where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. … ►If $g_j$ is the Laplace transform of $f_j$, $j=1,2$, then $g_1{g}_2$ is the Laplace transform of the convolution $f_1\ast f_2$, where …

25: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). … ►Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and ${𝐷}_n[a_{j,k}]$ denote the $(2n+1)\times (2n+1)$ determinant … ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. … ►The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$. …

26: Bibliography D

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On the zeros of generalized Bessel polynomials. I. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 1–13.

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ISSN 0019-3577, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.49(ii), §10.58.

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On the zeros of generalized Bessel polynomials. II. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 14–25.

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ISSN 0019-3577, Document, MathReview (M. E. Muldoon), ZentralBlatt

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§10.49(ii), §10.58.

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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Document

Referenced by:

§10.76(iii).

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Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

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Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

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27: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. The reduced mass is $m=m_1{m}_2/(m_1+m_2)$, and at energy of relative motion $E$ with relative orbital angular momentum $\mathrm{\ell }\mathrm{\hslash }$, the Schrödinger equation for the radial wave function $w(s)$ is given by … ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho /(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))$ (Mott and Massey (1956, pp. 63–65)). …

28: Bibliography O

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Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Online fulltext contains postpublication annotations.

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FullText

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§27.18.

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Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.

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Includes single, double precision Fortran code.

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GAMS (module DXNRMP; package SLATEC), ISSN 0021-9991, Document, MathReview (G. P. Bhattacharjee), ZentralBlatt

Referenced by:

§14.32, 7th item.

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Numerical solution of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 111–129.

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MathReview (G. N. Lance), ZentralBlatt

Referenced by:

§3.6(iv), §3.6(v).

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Error Analysis of Complex Arithmetic. In Computational Aspects of Complex Analysis (Braunlage, 1982), H. Werner, L. Wuytack, E. Ng, and H. J. Bünger (Eds.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 102, pp. 279–292.

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MathReview (G. Blanch), ZentralBlatt

Referenced by:

§3.1(v).

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Spectral decomposition of tent maps using symmetry considerations. J. Statist. Phys. 84 (1-2), pp. 269–276.

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ISSN 0022-4715, Document, MathReview (Makoto Mori), ZentralBlatt

Referenced by:

§24.18.

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29: 27.18 Methods of Computation: Primes

… ►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of $n-1,n+1$, or both; see Crandall and Pomerance (2005, §§4.1–4.2). These algorithms are used for testing primality of Mersenne numbers, $2^n-1$, and Fermat numbers, $2^{2^n}+1$. … ►The ECPP (Elliptic Curve Primality Proving) algorithm handles primes with over 20,000 digits. …

30: 18.39 Applications in the Physical Sciences

… ►where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\mathrm{\hslash }=h/(2\pi )$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval. … ►and $\mathrm{\hslash }=k=m=1$, has eigenfunctions … ►The eigenfunctions of ${\mathrm{L}}^2$ are the spherical harmonics $Y_{l,m_l}(\theta ,\varphi )$ with eigenvalues ${\mathrm{\hslash }}^{2}l(l+1)$, each with degeneracy $2l+1$ as $m_l=-l,-l+1,\mathrm{\dots },l$. … ►The eigenvalues and radial wave functions are independent of $m_l$ and they both do depend on $l$ due to the presence of the ‘fictitious’ centrifugal potential ${\mathrm{\hslash }}^{2}l(l+1)/(2m{r}^2)$, which is a result of the choice of co-ordinate system, and not the physical potential energy of interaction $V(r)$. … ►, $\mathrm{\hslash }=m_e=e^2=4\pi {ϵ}_0=1$, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of $a.u.$ to SI units is spelled out. …