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1: 11.6 Asymptotic Expansions

… ►where

\delta

is an arbitrary small positive constant. … ►

§11.6(ii) Large $|\nu|$ , Fixed $z$

►

11.6.5

\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},

|\operatorname{ph}\nu|\leq\pi-\delta

ⓘ

Symbols:: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order and $\delta$ : arbitrary small positive constant
Referenced by:: (11.11.7)
Permalink:: http://dlmf.nist.gov/11.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(ii), §11.6 and Ch.11

… ►

c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

… ►

11.6.9

\mathbf{L}_{\nu}\left(\lambda\nu\right)\sim I_{\nu}\left(\lambda\nu\right),

|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $\nu$ : real or complex order, $\delta$ : arbitrary small positive constant and $\lambda$ : parameter
Referenced by:: §11.6(iii)
Permalink:: http://dlmf.nist.gov/11.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(iii), §11.6 and Ch.11

…

2: Bibliography O

… ►

A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).
Encodings:: BibTeX

… ►

J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.

ⓘ

External Links:: Document, MathReview, ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1977b) 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.

ⓘ

External Links:: Document, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

►

F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

ⓘ

External Links:: FullText, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

…

3: Bibliography S

… ►

J. Shapiro (1970) Arbitrary

3n-j

symbols for

\mathrm{SU}(2)

. Comput. Phys. Comm. 1 (3), pp. 207–215.

ⓘ

Notes:: Single-precision Fortran
External Links:: Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

A. Sidi (1997) Computation of infinite integrals involving Bessel functions of arbitrary order by the

\overline{D}

-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

A. Sidi (2012a) Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp. 81 (280), pp. 2159–2173.

ⓘ

External Links:: ISSN 0025-5718, Document, Link, MathReview (Vittoria Demichelis), ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

►

A. Sidi (2012b) Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities. Constr. Approx. 36 (3), pp. 331–352.

ⓘ

External Links:: ISSN 0176-4276, Document, Link, MathReview (Razvan A. Mezei), ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

…

4: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. … ►The fundamental quantum Schrödinger operator, also called the Hamiltonian,

\mathcal{H}

, is a second order differential operator of the form … ►If

\Psi(x,t=0)=\chi(x)

is an arbitrary unit normalized function in the domain of

\mathcal{H}

then, by self-adjointness, … ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials

\mathbf{L}_{n+l}^{2l+1}(\rho_{n})

. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

5: 3.8 Nonlinear Equations

… ►for all

n

sufficiently large, where

A

and

p

are independent of

n

, then the sequence is said to have convergence of the

p

th order. … … ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. … ►For an arbitrary starting point

z_{0}\in\mathbb{C}

, convergence cannot be predicted, and the boundary of the set of points

z_{0}

that generate a sequence converging to a particular zero has a very complicated structure. …

6: Bibliography F

… ►

FDLIBM (free C library)

ⓘ

Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

… ►

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

… ►

A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §5.24(ii).
Encodings:: BibTeX

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

…

7: 9.9 Zeros

… ►They are denoted by

a_{k}

a^{\prime}_{k}

b_{k}

b^{\prime}_{k}

, respectively, arranged in ascending order of absolute value for

k=1,2,\ldots.

… ►They lie in the sectors

\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi

and

-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi

, and are denoted by

\beta_{k}

\beta^{\prime}_{k}

, respectively, in the former sector, and by

\overline{\beta_{k}}

\overline{\beta^{\prime}_{k}}

, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for

k=1,2,\ldots.

See §9.3(ii) for visualizations. ►For the distribution in

\mathbb{C}

of the zeros of

\operatorname{Ai}'\left(z\right)-\sigma\operatorname{Ai}\left(z\right)

, where

\sigma

is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … ►

9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.7

\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

…

8: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
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

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

… ►

B. T. M. Murphy and A. D. Wood (1997) Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. Methods Appl. Anal. 4 (3), pp. 250–260.

ⓘ

External Links:: ISSN 1073-2772, MathReview (W. Balser), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

…