Bibliography Bibliography U Bibliography W

Bibliography V

• G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  ⓘ

  External Links:

  ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

  Referenced by:

  §31.10(i).

  Encodings:

  BibTeX

• O. Vallée and M. Soares (2010) Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.
  ⓘ

  External Links:

  ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

  Referenced by:

  §1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

  Encodings:

  BibTeX

• A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.
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  External Links:

  Document, ZentralBlatt

  Referenced by:

  §30.16(ii), §30.18(iii).

  Encodings:

  BibTeX

• A. L. Van Buren, R. V. Baier, and S. Hanish (1970) A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..
  ⓘ

  Referenced by:

  §30.18(iii).

  Encodings:

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• A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish (1975) Tables of Angular Spheroidal Wave Functions, Vol. 1, Prolate, $m=0$; Vol. 2, Oblate, m=0. Naval Res. Lab. Reports, Washington, D.C..
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  Referenced by:

  5th item.

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• A. L. Van Buren and J. E. Boisvert (2002) Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.
  ⓘ

  Notes:

  For software, see Van Buren (website)

  External Links:

  ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt

  Referenced by:

  §30.16(iii).

  Encodings:

  BibTeX

• A. L. Van Buren and J. E. Boisvert (2004) Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Quart. Appl. Math. 62 (3), pp. 493–507.
  ⓘ

  Notes:

  For software, see Van Buren (website)

  External Links:

  ISSN 0033-569X, FullText, MathReview (Javier Segura), ZentralBlatt

  Referenced by:

  §30.16(iii), §30.16(iii).

  Encodings:

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• A. L. Van Buren and J. E. Boisvert (2007) Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.
  ⓘ

  Notes:

  For software, see Van Buren (website)

  External Links:

  ISSN 0033-569X, MathReview (Javier Segura), ZentralBlatt

  Referenced by:

  §28.36(iii).

  Encodings:

  BibTeX

• Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
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  External Links:

  Home Page

  Referenced by:

  § ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).

  Encodings:

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• H. C. van de Hulst (1957) Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.
  ⓘ

  External Links:

  MathReview (P. M. Morse)

  Referenced by:

  §9.16.

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• H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.
  ⓘ

  Referenced by:

  §6.17.

  Encodings:

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• J. van de Lune, H. J. J. te Riele, and D. T. Winter (1986) On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.
  ⓘ

  External Links:

  ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt

  Referenced by:

  §25.10(ii), §25.18(ii).

  Encodings:

  BibTeX

• H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
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  External Links:

  FullText, MathReview, ZentralBlatt

  Referenced by:

  §19.12, §19.5.

  Encodings:

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• C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  ⓘ

  Notes:

  A reprint was published in 1986.

  External Links:

  ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt

  Referenced by:

  §5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).

  Encodings:

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• A. J. van der Poorten (1980) Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  ⓘ

  External Links:

  MathReview (F. Beukers), ZentralBlatt

  Referenced by:

  (25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

  Encodings:

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• B. L. van der Waerden (1951) On the method of saddle points. Appl. Sci. Research B. 2, pp. 33–45.
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  External Links:

  Document, MathReview (E. T. Copson), ZentralBlatt

  Referenced by:

  §2.4(vi), §7.20(i).

  Encodings:

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• J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.
  ⓘ

  Notes:

  Associated computer program available online

  External Links:

  ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

  Referenced by:

  §10.74(vii), §10.74(vii).

  Encodings:

  BibTeX

• J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.
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  External Links:

  ISSN 0377-9017, Document, MathReview (Laurent Habsieger), ZentralBlatt

  Referenced by:

  §19.35(i).

  Encodings:

  BibTeX

• C. Van Loan (1992) Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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  External Links:

  ISBN 0-89871-285-8, MathReview (S. Hitotumatu), ZentralBlatt

  Referenced by:

  §3.11(v).

  Encodings:

  BibTeX

• B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.
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  Document

  Referenced by:

  §14.31(i).

  Encodings:

  BibTeX

• A. van Wijngaarden (1953) On the coefficients of the modular invariant $J(\tau )$. Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.
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  External Links:

  MathReview (J. Lehner), ZentralBlatt

  Referenced by:

  §23.17(ii).

  Encodings:

  BibTeX

• A. N. Varčenko (1976) Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).
  ⓘ

  Notes:

  English translation in Functional Anal. Appl. 18 (1976), pp. 175-196.

  External Links:

  ISSN 0374-1990, Document, MathReview (Helmut Hamm), ZentralBlatt

  Referenced by:

  §36.6.

  Encodings:

  BibTeX

• R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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  External Links:

  MathReview (M. C. Gray), ZentralBlatt

  Referenced by:

  §12.13(ii).

  Encodings:

  BibTeX

• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ (1988) Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Singapore.
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  External Links:

  ISBN 9971-50-107-4, MathReview (M. A. Rashid), ZentralBlatt

  Referenced by:

  §16.24(iii), §34.1, §34.11, §34.13, §34.13, §34.14, §34.2, §34.3(iii), §34.3(vi), §34.4, §34.4, §34.5(iii), §34.5(vi), §34.7(ii), §34.7(iii), §34.8, §34.9, Ch.34.

  Encodings:

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• A. N. Vavreck and W. Thompson (1984) Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.
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  External Links:

  ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt

  Referenced by:

  §10.60(iii), §10.60(iii).

  Encodings:

  BibTeX

• G. Vedeler (1950) A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.
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  External Links:

  MathReview (J. V. Wehausen), ZentralBlatt

  Referenced by:

  2nd item.

  Encodings:

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• R. Vein and P. Dale (1999) Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York.
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  External Links:

  ISBN 0-387-98558-1, MathReview (Jorma Kaarlo Merikoski), ZentralBlatt

  Referenced by:

  §1.3(ii).

  Encodings:

  BibTeX

• G. Veneziano (1968) Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A 57 (1), pp. 190–197.
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  FullText

  Referenced by:

  §5.20.

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• P. Verbeeck (1970) Rational approximations for exponential integrals $E_n(x)$. Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.
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  External Links:

  MathReview (A. Magnus), ZentralBlatt

  Referenced by:

  3rd item.

  Encodings:

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• A. Verma and V. K. Jain (1983) Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
  ⓘ

  External Links:

  ISSN 0019-5839, MathReview Entry, ZentralBlatt

  Referenced by:

  (17.6.4_5), §17.6(i), (17.8.8), §17.8.

  Encodings:

  BibTeX

• R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.
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  External Links:

  ISSN 0022-247X, Document, MathReview, ZentralBlatt

  Referenced by:

  §2.4(v).

  Encodings:

  BibTeX

• R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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  External Links:

  ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

  Referenced by:

  §15.8(v).

  Encodings:

  BibTeX

• L. Vietoris (1983) Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).
  ⓘ

  External Links:

  ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt

  Referenced by:

  §8.10.

  Encodings:

  BibTeX

• N. Ja. Vilenkin and A. U. Klimyk (1991) Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.
  ⓘ

  Notes:

  Translated from the Russian by V. A. Groza and A. A. Groza

  External Links:

  ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

  Referenced by:

  §18.38(iii).

  Encodings:

  BibTeX

• N. Ja. Vilenkin and A. U. Klimyk (1992) Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
  ⓘ

  Notes:

  Translated from the Russian by V. A. Groza and A. A. Groza

  External Links:

  ISBN 0-7923-1493-X, MathReview (P. D. F. Ion), ZentralBlatt

  Referenced by:

  §18.38(iii), Ch.35.

  Encodings:

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• N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
  ⓘ

  Notes:

  Translated from the Russian by V. A. Groza and A. A. Groza

  External Links:

  ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

  Referenced by:

  §18.38(iii).

  Encodings:

  BibTeX

• N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
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  External Links:

  MathReview, ZentralBlatt

  Referenced by:

  §13.27.

  Encodings:

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• I. M. Vinogradov (1937) Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).
  ⓘ

  External Links:

  ZentralBlatt

  Referenced by:

  §27.13(ii).

  Encodings:

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• I. M. Vinogradov (1958) A new estimate of the function $\zeta (1+it)$. Izv. Akad. Nauk SSSR. Ser. Mat. 22, pp. 161–164 (Russian).
  ⓘ

  External Links:

  MathReview (H. Davenport), ZentralBlatt

  Referenced by:

  §27.12.

  Encodings:

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• N. Virchenko and I. Fedotova (2001) Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.
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  External Links:

  ISBN 981-02-4353-7, MathReview (B. D. Agrawal), ZentralBlatt

  Referenced by:

  §14.29.

  Encodings:

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• H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
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  External Links:

  ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

  Referenced by:

  §31.15(iii).

  Encodings:

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• H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.
  ⓘ

  External Links:

  ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt

  Referenced by:

  §16.11(ii), §16.11(ii).

  Encodings:

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• H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
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  External Links:

  ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

  Referenced by:

  §29.8, §29.8.

  Encodings:

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• H. Volkmer (1983) Integralgleichungen für periodische Lösungen Hill’scher Differentialgleichungen. Analysis 3 (1-4), pp. 189–203 (German).
  ⓘ

  External Links:

  ISSN 0174-4747, MathReview (H. Hochstadt), ZentralBlatt

  Referenced by:

  §28.10(iii), §29.8, §29.8.

  Encodings:

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• H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
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  External Links:

  ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

  Referenced by:

  §29.8.

  Encodings:

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• H. Volkmer (1998) On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.
  ⓘ

  External Links:

  ISSN 0025-584X, Document, MathReview (M. E. Muldoon), ZentralBlatt

  Referenced by:

  §28.6(i).

  Encodings:

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• H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  ⓘ

  External Links:

  ISSN 0176-4276, Document, MathReview, ZentralBlatt

  Referenced by:

  §30.16(i), §30.16(ii), §30.16(ii).

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• H. Volkmer (2004b) Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.
  ⓘ

  External Links:

  ISSN 0219-5305, Document, MathReview, ZentralBlatt

  Referenced by:

  §29.3(ii), §29.3(vii), §29.5, §29.7(i), §29.7(i).

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• H. Volkmer (2008) Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.
  ⓘ

  External Links:

  ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

  Referenced by:

  item (f), §28.34(ii), §29.20(i).

  Encodings:

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• H. Volkmer (2021) Fourier series representation of Ferrers function $𝖯$.
  ⓘ

  External Links:

  FullText

  Referenced by:

  (14.13.1).

  Encodings:

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• H. Volkmer (2023) Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.
  ⓘ

  External Links:

  ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

  Referenced by:

  §16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

  Encodings:

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• H. von Koch (1901) Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).
  ⓘ

  External Links:

  ISSN 0025-5831; 1432-1807/e, Document, MathReview Entry, ZentralBlatt

  Referenced by:

  §27.12.

  Encodings:

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• A. P. Vorob’ev (1965) On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).
  ⓘ

  Notes:

  In Russian. English translation: Differential Equations 1(1), pp. 58–59.

  External Links:

  MathReview, ZentralBlatt

  Referenced by:

  §32.8(ii).

  Encodings:

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• M. N. Vrahatis, T. N. Grapsa, O. Ragos, and F. A. Zafiropoulos (1997a) On the localization and computation of zeros of Bessel functions. Z. Angew. Math. Mech. 77 (6), pp. 467–475.
  ⓘ

  External Links:

  ISSN 0044-2267, Document, MathReview, ZentralBlatt

  Referenced by:

  §10.74(vi).

  Encodings:

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• M. N. Vrahatis, O. Ragos, T. Skiniotis, F. A. Zafiropoulos, and T. N. Grapsa (1995) RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Comm. 92 (2-3), pp. 252–266.
  ⓘ

  Notes:

  Double-precision Fortran, maximum accuracy 16S. For corrections see same journal 117 (1999), p. 290

  External Links:

  ISSN 0010-4655, Document, Code, ZentralBlatt

  Referenced by:

  3rd item.

  Encodings:

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• M. N. Vrahatis, O. Ragos, T. Skiniotis, F. A. Zafiropoulos, and T. N. Grapsa (1997b) The topological degree theory for the localization and computation of complex zeros of Bessel functions. Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.
  ⓘ

  External Links:

  ISSN 0163-0563, Document, MathReview, ZentralBlatt

  Referenced by:

  §10.74(vi).

  Encodings:

  BibTeX

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