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1: Bibliography D
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • 2: 12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10(vi) Modifications of Expansions in Elementary Functions
    Modified Expansions
    3: Bibliography I
  • J. Igusa (1972) Theta Functions. Springer-Verlag, New York.
  • Y. Ikebe, Y. Kikuchi, and I. Fujishiro (1991) Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.
  • E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.
  • E. L. Ince (1940a) The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • 4: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.
  • A. Gil, J. Segura, and N. M. Temme (2006a) Computing the real parabolic cylinder functions U ( a , x ) , V ( a , x ) . ACM Trans. Math. Software 32 (1), pp. 70–101.
  • A. Gil, J. Segura, and N. M. Temme (2006b) Algorithm 850: Real parabolic cylinder functions U ( a , x ) , V ( a , x ) . ACM Trans. Math. Software 32 (1), pp. 102–112.
  • A. Gil, J. Segura, and N. M. Temme (2011a) Algorithm 914: parabolic cylinder function W ( a , x ) and its derivative. ACM Trans. Math. Software 38 (1), pp. Art. 6, 5.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • 5: 12.19 Tables
    §12.19 Tables
  • Miller (1955) includes W ( a , x ) , W ( a , x ) , and reduced derivatives for a = 10 ( 1 ) 10 , x = 0 ( .1 ) 10 , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

  • Fox (1960) includes modulus and phase functions for W ( a , x ) and W ( a , x ) , and several auxiliary functions for x 1 = 0 ( .005 ) 0.1 , a = 10 ( 1 ) 10 , 8S.

  • Murzewski and Sowa (1972) includes D n ( x ) ( = U ( n 1 2 , x ) ) for n = 1 ( 1 ) 20 , x = 0 ( .05 ) 3 , 7S.

  • 6: Bibliography S
  • J. Segura and A. Gil (1999) Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.
  • J. Segura (1998) A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.
  • H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • B. D. Sleeman (1968b) On parabolic cylinder functions. J. Inst. Math. Appl. 4 (1), pp. 106–112.
  • 7: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • C. L. Adler, J. A. Lock, B. R. Stone, and C. J. Garcia (1997) High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder. J. Opt. Soc. Amer. A 14 (6), pp. 1305–1315.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • 8: Software Index
    Open Source With Book Commercial
    20 Theta Functions
    ‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    9: Bibliography F
  • M. Faierman (1992) Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.
  • FDLIBM (free C library)
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
  • G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
  • 10: Bibliography C
  • B. C. Carlson (1985) The hypergeometric function and the R -function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
  • 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.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.