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11: 10.72 Mathematical Applications

… ►The number $m$ can also be replaced by any real constant $\lambda$ $(>-2)$ in the sense that ${(z-z_0)}^{-\lambda }$ $f(z)$ is analytic and nonvanishing at $z_0$; moreover, $g(z)$ is permitted to have a single or double pole at $z_0$. …

12: Bibliography N

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Numerical Algorithms Group, Ltd..

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Notes:

The NAG Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Notes:

Single and double precision. See Kahaner et al. (1989).

External Links:

GAMS (package NMS)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Notes:

A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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13: 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 ${ϵ}_n$, $n\ge 0$. … ►The non-relativistic Schrödinger equation describing a single, bound (negative energy) electron, in an $L^2$ eigenstate of energy $E$ is: … ►Namely for fixed $l$ the infinite set labeled by $p$ describe only the $L^2$ bound states for that single $l$, omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii). … ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. … ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …

14: Bibliography T

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Coulomb functions with complex angular momenta. Comput. Phys. Comm. 17 (4), pp. 351–355.

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Notes:

Single-precision Fortran code.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

3rd item, 2nd item.

Encodings:

BibTeX

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

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Angular momentum coupling coefficients. Comput. Phys. Comm. 1 (5), pp. 337–342.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

11st item.

Encodings:

BibTeX

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15: 20.5 Infinite Products and Related Results

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§20.5(i) Single Products

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16: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►Alternative notations are $\mathrm{Wp}\left(x\right)$ for $W_0\left(x\right)$, $\mathrm{Wm}\left(x\right)$ for $W_{-1}\left(x+0\mathrm{i}\right)$, both previously used in this section, the Wright $\omega$-function $\omega \left(z\right)=W\left({\mathrm{e}}^z\right)$, which is single-valued, satisfies …

17: Bibliography F

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

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

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

BibTeX

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Notes:

This library treats more than 100 mathematical functions. Implementations in single and double precision are provided. See Fullerton (1977).

External Links:

GAMS (package FN)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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The PORT mathematical subroutine library. ACM Trans. Math. Software 4 (2), pp. 104–126.

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Notes:

Single- and double-precision Fortran.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ii), §10.77(iv).

Encodings:

BibTeX

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18: 20.1 Special Notation

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$m$, $n$	integers.
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$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
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19: 22.18 Mathematical Applications

… ►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. …

20: 36.7 Zeros

… ►Just outside the cusp, that is, for $x^2>8{|y|}^3/27$, there is a single row of zeros on each side. …