relation to elementary functions

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11: 3.11 Approximation Techniques

… ►For examples of minimax polynomial approximations to elementary and special functions see Hart et al. (1968). … ►They satisfy the recurrence relation … ► … ►Also, in cases where $f(x)$ satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the $c_n$. … ►A collection of minimax rational approximations to elementary and special functions can be found in Hart et al. (1968). …

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► The analogous orthonormality is … ►and completeness relation … ► … ►Then orthogonality and normalization relations are …The formal completeness relation is now …

13: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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MathReview (A. Erdélyi), ZentralBlatt

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Ch.14.

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On the interval computation of elementary functions. C. R. Acad. Bulgare Sci. 34 (3), pp. 319–322.

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ISSN 0366-8681, MathReview (David F. Griffiths), ZentralBlatt

Referenced by:

§4.45(i).

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The computation of elementary functions in radix $2^p$ . Computing 53 (3-4), pp. 219–232.

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International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Vienna, 1993)

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ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i).

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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

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§ ‣ 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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Elementary Functions: Algorithms and Implementation. Birkhäuser Boston Inc., Boston, MA.

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Notes:

A 2nd edition was published in 2006.

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ISBN 0-8176-3990-X, MathReview (Warren E. Ferguson, Jr.), ZentralBlatt

Referenced by:

§4.45(i).

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14: Notices

… ►Authors of the works appearing in the Digital Library of Mathematical Functions (DLMF) have assigned copyright to the works to NIST, United States Department of Commerce, as represented by the Secretary of Commerce. … ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …Thus, we seek to provide DLMF users with links to sources of such software. … ►

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

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In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

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15: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). ►For related results see Langer (1934) and Sharples (1967, 1971). …

16: 1.11 Zeros of Polynomials

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§1.11(ii) Elementary Properties

… ►A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … ►The elementary symmetric functions of the zeros are (with $a_n\ne 0$) … ►Addition of $-\frac{1}{3}a$ to each of these roots gives the roots of $f(z)=0$. … ►Add $-\frac{1}{4}a$ to the roots of $g(w)=0$ to get those of $f(z)=0$. …

17: 18.15 Asymptotic Approximations

… ►For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). … ►

In Terms of Elementary Functions

… ►For more powerful asymptotic expansions as $n\to \mathrm{\infty }$ in terms of elementary functions that apply uniformly when , $-1+\delta \le t\le 1-\delta$, or , where $t=x/\sqrt{2n+1}$ and $\delta$ is again an arbitrary small positive constant, see §§12.10(i)–12.10(iv) and 12.10(vi). And for asymptotic expansions as $n\to \mathrm{\infty }$ in terms of Airy functions that apply uniformly when or , see §§12.10(vii) and 12.10(viii). With $\mu =\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, which is related to the Hermite polynomials via …

18: Bibliography S

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Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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FullText, Document, MathReview (L. Fox), ZentralBlatt

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3rd item.

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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External Links:

ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

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§31.13.

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

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Efficient multiple-precision evaluation of elementary functions. Math. Comp. 52 (185), pp. 131–134.

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External Links:

ISSN 0025-5718, FullText, MathReview (Menachem Dishon), ZentralBlatt

Referenced by:

§4.45(i).

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19: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

… ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►For elementary functions, see §§ 4.9 and 4.35. … ►This forward algorithm achieves efficiency and stability in the computation of the convergents $C_n=A_n/B_n$, and is related to the forward series recurrence algorithm. …

20: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …where the elementary symmetric functions $E_s$ are defined by (19.19.4). … ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►To (19.36.6) add … ►

§19.36(iii) Via Theta Functions

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