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21: Bibliography Y

… ►

K. Yang and M. de Llano (1989) Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

ⓘ

External Links:: Document
Referenced by:: §1.18(viii).
Encodings:: BibTeX

…

22: 1.16 Distributions

… ►

§1.16(i) Test Functions

… ►The closure of the set of points where

\phi\not=0

is called the support of

\phi

. If the support of

\phi

is a compact set (§1.9(vii)), then

\phi

is called a function of compact support. A test function is an infinitely differentiable function of compact support. …

A. Máté, P. Nevai, and W. Van Assche (1991) The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

ⓘ

External Links:: ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)
Referenced by:: §18.2(xi).
Encodings:: BibTeX

… ►

Maxima (free interactive system)

ⓘ

Notes:: System for the manipulation of symbolic and numerical expressions. Includes a collection of special functions. Utilizes exact fractions, arbitrary precision integers, and arbitrary precision floating point numbers.
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.
Encodings:: BibTeX

… ►

S. L. B. Moshier (1989) Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.

ⓘ

Notes:: Includes diskette
External Links:: ISBN 0-13-578998-2, MathReview, ZentralBlatt
Referenced by:: 2nd item, CEPHES (free C library).
Encodings:: BibTeX

… ►

MPFR (free C library)

ⓘ

Notes:: MPFR is a C library for multiple-precision floating-point computations with correct rounding. The main goal of MPFR is to provide a library for multiple-precision floating-point computation which is both efficient and has well-defined semantics. MPFR includes most of the elementary functions. It also implements the gamma function, and more special functions are planned in the future.
External Links:: MPFR
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.
Encodings:: BibTeX

►

mpmath (free python library)

ⓘ

Notes:: Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.
External Links:: mpmath
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, § ‣ 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, § ‣ 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, § ‣ 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

…

24: 18.2 General Orthogonal Polynomials

… ►The moments for an orthogonality measure

\,\mathrm{d}\mu(x)

are the numbers … ►are the Christoffel numbers, see also (3.5.18). … ►Nevai (1979, p.39) defined the class

\mathcal{S}

of orthogonality measures with support inside

[-1,1]

such that the absolutely continuous part

w(x)\,\mathrm{d}x

has

w

in the Szegő class

\mathcal{G}

. … ►If

\,\mathrm{d}\mu\in{\mathbf{M}}(a,b)

then the interval

[b-a,b+a]

is included in the support of

\,\mathrm{d}\mu

, and outside

[b-a,b+a]

the measure

\,\mathrm{d}\mu

only has discrete mass points

x_{k}

such that

b\pm a

are the only possible limit points of the sequence

\{x_{k}\}

, see Máté et al. (1991, Theorem 10). … ►for

x, y

in the support of the orthogonality measure and

z

such that the series in (18.2.41) converges absolutely for all these

x, y

. …

25: Bibliography K

… ►

D. K. Kahaner, C. Moler, and S. Nash (1989) Numerical Methods and Software. Prentice Hall, Englewood Cliffs, N.J..

ⓘ

Notes:: Includes collection of single- and double-precision Fortran subroutines.
External Links:: ISBN 0-13-627258-4, GAMS (package NMS), ZentralBlatt
Referenced by:: NMS (free collection of Fortran subroutines).
Encodings:: BibTeX

… ►

M. Kaneko (1997) Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

ⓘ

External Links:: ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.15(iv).
Encodings:: BibTeX

… ►

T. Kim and H. S. Kim (1999) Remark on

p

-adic

q

-Bernoulli numbers. Adv. Stud. Contemp. Math. (Pusan) 1, pp. 127–136.

ⓘ

Notes:: Algebraic number theory (Hapcheon/Saga, 1996)
External Links:: ISSN 1229-3067, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

C. Kormanyos (2011) Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.

ⓘ

Notes:: This system provides a uniform interface in C++, named e_float, to perform arithmetic operations and to calculate mathematical functions that are implemented in several different MP packages. It is a free, open-source, multiple precision package that allows the computation of many special functions. It supports calculations with 30….300 decimal digits and is interoperable with Microsoft’s CLR, Python and Mathematica.
External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: Erratum (V1.0.9) for References, § ‣ 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, § ‣ 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

…

26: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Let

\mu

be a probability measure on the unit circle of which the support is an infinite set. … ►This states that for any sequence

\{\alpha_{n}\}_{n=0}^{\infty}

with

\alpha_{n}\in\mathbb{C}

and

|\alpha_{n}|<1

the polynomials

\Phi_{n}(z)

generated by the recurrence relations (18.33.23), (18.33.24) with

\Phi_{0}(z)=1

satisfy the orthogonality relation (18.33.17) for a unique probability measure

\mu

with infinite support on the unit circle. …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Conversely, if complex numbers

c_{n}

satisfy (1.18.5) then there is a unique

v\in V

such that (1.18.3) holds and

v

can be given by … ►For

\mathcal{D}(T)

we can take

C^{2}(X)

, with appropriate boundary conditions, and with compact support if

X

is bounded, which space is dense in

L^{2}\left(X\right)

, and for

X

unbounded require that possible non- $L^{2}$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including

\pm\infty

. … ►The number,

N

, of discrete states depends on the nature of

V(r)

, as well as

\ell

, and, again,

V(r)

must vanish as

r\to\infty

, corresponding to the traditionally assumed start of the energy continuum at

\lambda=0

. …

28: Errata

… ►This release increments the minor version number and contains considerable additions of new material and clarifications. … ►

Native MathML

DLMF now uses browser-native MathML rendering for mathematics, by default, in all browsers which support MathML. See About MathML for more details and for other options.

… ►This release increments the minor version number and contains considerable additions of new material and clarifications. These additions were facilitated by an extension of the scheme for reference numbers; with “_” introducing intermediate numbers. … ►

Usability

In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

…

29: Preface

… ►The new DLMF (Digital Library of Mathematical Functions) will appear in a hardcover edition and as a free electronic publication on the World Wide Web. The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. …

30: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with

N+1

regular singularities at

z=a_{j}

j=1,2,\dots,N

, and at

\infty

, is given by ►

31.14.1

{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},

\sum_{j=1}^{N}q_{j}=0

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14 and Ch.31

… ►The three sets of parameters comprise the singularity parameters

a_{j}

, the exponent parameters

\alpha,\beta,\gamma_{j}

, and the

N-2

free accessory parameters

q_{j}

. With

a_{1}=0

and

a_{2}=1

the total number of free parameters is

3N-3

. … ►

31.14.3

w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

…