completely multiplicative functions

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11: Bibliography M

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Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

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Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.

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Notes:

Includes Algol program, maximum accuracy 10D. For remark see ACM Trans. Math. Software 4 (1978), p. 95

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

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On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.

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External Links:

Document, MathReview (V. F. Cowling), ZentralBlatt

Referenced by:

(25.6.5).

Encodings:

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Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.

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Notes:

An Algol procedure for calculating the complete elliptic integrals of the first and the second kind with complex modulus $k$ is included.

External Links:

ISSN 0029-599X, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.39(ii).

Encodings:

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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.

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12: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}}$$

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

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The multi-product notation ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

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17.2.22 $$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a}$$

17.2.23 $$\frac{{(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}{{(a^{\frac{1}{k}},{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}=\frac{{(a{q}^k;q^k)}_n}{{(a;q^k)}_n}=\frac{1-a{q}^{kn}}{1-a}$$

The numerators of the leftmost fractions were corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ and ${(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$ and ${(a{q}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$, respectively.

Reported 2017-06-26 by Jason Zhao.

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An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).

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13: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►

§18.36(iii) Multiple Orthogonal Polynomials

… ►This infinite set of polynomials of order $n\ge k$, the smallest power of $x$ being $x^k$ in each polynomial, is a complete orthogonal set with respect to this measure. … ►Completeness follows from the self-adjointness of $T_k$, Everitt (2008). … ►Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).

14: 3.11 Approximation Techniques

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Example

… ►The rational function … ►requires approximately $n^2$ multiplications. … ►For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating $f(x)$ on the complete interval $[a,b]$. … ►A complete spline results by composing several Bézier curves. …

15: Bibliography

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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

Encodings:

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Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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External Links:

ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§19.9(i).

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Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.9(i).

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

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16: Bibliography O

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

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Triorthogonal systems in spaces of constant curvature in which the equation ${\mathrm{\Delta }}_{2}u+\lambda u=0$ allows a complete separation of variables. Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).

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Notes:

In Russian.

External Links:

MathReview (H. Busemann), ZentralBlatt

Referenced by:

§31.17(i).

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

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17: Bibliography B

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Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12 (4), pp. 519–526.

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External Links:

ISSN 0272-4979, MathReview, ZentralBlatt

Referenced by:

§5.21, §5.24(ii).

Encodings:

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A Fortran multiple-precision arithmetic package. ACM Trans. Math. Software 4 (1), pp. 57–70.

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External Links:

ISSN 0098-3500, Document

Referenced by:

1st item.

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Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Mach. 23 (2), pp. 242–251.

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External Links:

Document, MathReview (Amnon Barak), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

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Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

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Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

External Links:

ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

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Numerical calculation of elliptic integrals and elliptic functions. II. Numer. Math. 7 (4), pp. 353–354.

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Notes:

An improved Algol 60 program for the complete elliptic integral of the third kind is included, but superseded by a program in Bulirsch (1969b).

External Links:

ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§19.1.

Encodings:

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