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11: Bibliography H

… ►

J. R. Herndon (1961a) Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.

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Notes:: Includes Algol program, maximum accuracy 8D. For certification see same journal 6 (1966), pp. 166–167
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

… ►

D. R. Herrick and S. O’Connor (1998) Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

ⓘ

External Links:: Document
Referenced by:: §15.18.
Encodings:: BibTeX

… ►

H. W. Hethcote (1970) Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

… ►

G. W. Hill (1970) Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

ⓘ

Notes:: Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)
External Links:: ISSN 0001-0782, Document, Remark 1, Remark 2
Referenced by:: §8.28(iv).
Encodings:: BibTeX

… ►

C. Hunter (1981) Two Parametric Eigenvalue Problems of Differential Equations. In Spectral Theory of Differential Operators (Birmingham, AL, 1981), North-Holland Math. Stud., Vol. 55, pp. 233–241.

ⓘ

External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

…

… ►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …

13: Bibliography E

… ►

C. Eckart (1930) The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §15.18.
Encodings:: BibTeX

… ►

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953a) Higher Transcendental Functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.

ⓘ

Notes:: Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, p. 778, v. 30 (1976), no. 135, p. 675, v. 25 (1971), no. 115, p. 635, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 112, p. 999, v. 24 (1970), no. 110, p. 504, v. 17 (1963), no. 84, p. 485.
External Links:: MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §13.1, §13.10(ii), §13.10(iv), §13.12, §13.13(i), §13.16(ii), §13.16(iii), §13.23(i), §13.23(ii), §13.25, §13.4(i), §13.4(ii), §13.9(i), §13.9(ii), Ch.13, §14.1, §14.10, §14.12(ii), §14.12(i), §14.12(ii), §14.13, §14.13, §14.17(ii), §14.17(iv), §14.18(ii), §14.18(iii), §14.18(iv), §14.19(i), §14.19(ii), §14.19(iii), §14.19(iv), §14.20(v), §14.21(iii), §14.25, §14.28(i), §14.28(ii), §14.3(iii), §14.3(iii), §14.3(iv), §14.5(iii), §14.5(iv), §14.6(i), §14.6(ii), §14.7(iv), §14.8(i), §14.8(ii), Ch.14, §15.16, §15.4(iii), §15.5(i), §15.6, §15.8(v), §15.8(ii), §15.8(v), §15.9(iii), Ch.15, §16.12, §16.12, §16.13, §16.14(i), §16.14(ii), (16.15.3), §16.15, §16.16(i), §16.16(ii), §16.18, §16.20, §16.20, §16.21, §16.4(ii), §16.5, §16.6, Ch.16, §19.21(i), §19.28, §19.5, §24.16(iii), §24.5(i), §24.7(i), §24.7(ii), §24.8(i), Ch.24, (25.11.12), (25.11.18), (25.11.29), (25.11.30), (25.12.12), (25.12.13), (25.14.1), (25.14.4), (25.14.5), (25.14.6), §25.14(i), §25.14(i), §25.14(ii), §25.14(ii), §25.14, (25.5.1), (25.5.10), (25.5.11), (25.5.20), (25.5.21), (25.5.3), (25.5.8), (25.5.9), §25.5(i), §25.5(iii), §25.5, (25.8.5), (25.8.6), §25.8, Ch.25, §5.7(i), Ch.5, Notations P, Notations P, Notations Q.
Encodings:: BibTeX

►

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953b) Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

ⓘ

Notes:: Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 41 (1983), no. 164, p. 778, v. 36 (1981), no. 153, p. 315, v. 30 (1976), no. 135, p. 675, v. 29 (1975), no. 130, p. 670, v. 26 (1972), no. 118, p. 598, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239, v. 17 (1963), no. 84, p. 485.
External Links:: MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §10.22(iv), §10.22(vi), §10.23(iii), §10.23(iii), §10.23(iv), §10.32(i), §10.32(iii), §10.32(iv), §10.43(iv), §10.43(vi), §10.44(iv), §10.60(iv), §10.9(i), §10.9(iii), §10.9(iv), §11.10(vi), §11.4(i), Ch.11, §12.12, §12.12, §12.13(i), §12.5(iv), Ch.12, §18.16(vi), (18.17.21_1), §18.17(iv), §18.17(i), §18.17(viii), (18.35.1), (18.35.2), (18.35.4_5), §18.35(i), (18.5.11_1), (18.5.11_3), §18.5(ii), §18.9(iii), §19.1, §19.10(i), §19.14(ii), (19.25.40), §19.25(vi), §19.5, §19.7(ii), §19.9(i), Ch.19, §20.9(i), §22.15(ii), §23.6(iv), Ch.23, §8.13(i), §8.14, §8.3(i), §8.4, §8.6(iii), §8.7, Ch.8, Notations P, Notations P.
Encodings:: BibTeX

… ►

L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.

ⓘ

Referenced by:: §25.12(i).
Encodings:: BibTeX

►

G. A. Evans and J. R. Webster (1999) A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §3.5(vii).
Encodings:: BibTeX

…

14: Bibliography N

… ►

G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.16), (11.11.17), §11.11(iii).
Encodings:: BibTeX

… ►

E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

ⓘ

Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

►

E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

C. J. Noble (2004) Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

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Notes:: Double-precision Fortran-90 code.
External Links:: Document, Code
Referenced by:: §33.23(vi), 8th item.
Encodings:: BibTeX

… ►

V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

ⓘ

Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

…

15: Bibliography

… ►

M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:: MathReview (J. G. van der Corput), ZentralBlatt
Referenced by:: §30.9(iii).
Encodings:: BibTeX

… ►

S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

ⓘ

External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

… ►

F. Alhargan and S. Judah (1995) A general mode theory for the elliptic disk microstrip antenna. IEEE Trans. Antennas and Propagation 43 (6), pp. 560–568.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

… ►

R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

ⓘ

External Links:: ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.
Encodings:: BibTeX

…

16: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.2

\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),

|1-z|\leq 2

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\operatorname{arccos}\NVar{z}$ : arccosine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.)
Permalink:: http://dlmf.nist.gov/4.24.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.5

\operatorname{arctan}z=\frac{z}{z^{2}+1}\*\left(1+\frac{2}{3}\frac{z^{2}}{1+z^% {2}}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{1+z^{2}}\right)^{2}+\cdots% \right),

\Re\left(z^{2}\right)>-\tfrac{1}{2}

ⓘ

Symbols:: $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 4.4.42 (has an error in the conditions on $z$ .)
Permalink:: http://dlmf.nist.gov/4.24.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.6

x^{2}-y^{2}=-\tfrac{1}{2}.

ⓘ

Symbols:: $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/4.24.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.14

\operatorname{Arccos}u\pm\operatorname{Arccos}v=\operatorname{Arccos}\left(uv% \mp((1-u^{2})(1-v^{2}))^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccos}\NVar{z}$ : general arccosine function
A&S Ref:: 4.4.33
Permalink:: http://dlmf.nist.gov/4.24.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

… ►

4.24.16

\operatorname{Arcsin}u\pm\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv% \pm((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/% 2}\mp u(1-v^{2})^{1/2}\right),

ⓘ

Symbols:: $\operatorname{Arccos}\NVar{z}$ : general arccosine function and $\operatorname{Arcsin}\NVar{z}$ : general arcsine function
A&S Ref:: 4.4.35
Permalink:: http://dlmf.nist.gov/4.24.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(iii), §4.24 and Ch.4

…

17: 24.13 Integrals

… ►For

m,n=1,2,\dotsc

, … ►

24.13.9

\int_{0}^{1/2}E_{2n}\left(t\right)\,\mathrm{d}t=-\frac{E_{2n+1}\left(0\right)}% {2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

►

24.13.10

\int_{0}^{1/2}E_{2n-1}\left(t\right)\,\mathrm{d}t=\frac{E_{2n}}{n2^{2n+1}},

n=1,2,\dots

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Referenced by:: §24.13(ii)
Permalink:: http://dlmf.nist.gov/24.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

►For

m,n=1,2,\dotsc

, … ►For other integrals see Prudnikov et al. (1990, pp. 55–57).

18: Bibliography C

… ►

M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

J. N. L. Connor (1976) Catastrophes and molecular collisions. Molecular Phys. 31 (1), pp. 33–55.

ⓘ

External Links:: Document, MathReview (V. M. Zakaljukin)
Referenced by:: §36.14(iii).
Encodings:: BibTeX

… ►

F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

ⓘ

External Links:: ISSN 1076-2787, Document, MathReview
Referenced by:: §19.6(i).
Encodings:: BibTeX

… ►

A. Csótó and G. M. Hale (1997)

S

-matrix and

R

-matrix determination of the low-energy

{}^{5}\mathrm{He}

and

{}^{5}\mathrm{Li}

resonance parameters. Phys. Rev. C 55 (1), pp. 536–539.

ⓘ

External Links:: Document
Referenced by:: 2nd item.
Encodings:: BibTeX

…

19: Bibliography G

… ►

B. Gambier (1910) Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.

ⓘ

External Links:: ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt
Referenced by:: §32.10(ii), §32.10(iv), §32.7(ii).
Encodings:: BibTeX

… ►

W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2002c) Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §9.17(iii).
Encodings:: BibTeX

… ►

P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

ⓘ

External Links:: ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

…

20: 10.60 Sums

… ►

10.60.11

\sum_{n=0}^{\infty}{\mathsf{j}_{n}}^{2}\left(z\right)=\frac{\operatorname{Si}% \left(2z\right)}{2z}.

ⓘ

Symbols:: $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.52
Referenced by:: §10.60(iii), Ch.10
Permalink:: http://dlmf.nist.gov/10.60.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

… ►

10.60.12

\sum_{n=0}^{\infty}(2n+1){\mathsf{j}_{n}}^{2}\left(z\right)=1,

ⓘ

Symbols:: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.50
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

►

10.60.13

\sum_{n=0}^{\infty}(-1)^{n}(2n+1){\mathsf{j}_{n}}^{2}\left(z\right)=\frac{\sin% \left(2z\right)}{2z},

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable
A&S Ref:: 10.1.51
Referenced by:: §10.60(iii)
Permalink:: http://dlmf.nist.gov/10.60.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §10.60(iii), §10.60 and Ch.10

… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). … ►See also Watson (1944, Chapters 11 and 16).