About the Project
NIST

of arbitrary order

AdvancedHelp

(0.002 seconds)

11—20 of 47 matching pages

11: 14.24 Analytic Continuation
14.24.1 P ν - μ ( z e s π i ) = e s ν π i P ν - μ ( z ) + 2 i sin ( ( ν + 1 2 ) s π ) e - s π i / 2 cos ( ν π ) Γ ( μ - ν ) Q ν μ ( z ) ,
14.24.2 Q ν μ ( z e s π i ) = ( - 1 ) s e - s ν π i Q ν μ ( z ) ,
14.24.4 Q ν , s μ ( z ) = e - s μ π i Q ν μ ( z ) - π i sin ( s μ π ) sin ( μ π ) Γ ( ν - μ + 1 ) P ν - μ ( z ) ,
12: 11.1 Special Notation
§11.1 Special Notation
x

real variable.

ν

real or complex order.

n

integer order.

δ

arbitrary small positive constant.

13: 10.1 Special Notation
m , n

integers. In §§10.4710.71 n is nonnegative.

δ

arbitrary small positive constant.

For the spherical Bessel functions and modified spherical Bessel functions the order n is a nonnegative integer. For the other functions when the order ν is replaced by n , it can be any integer. For the Kelvin functions the order ν is always assumed to be real. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
14: Bibliography O
  • 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.
  • 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.
  • 15: 11.11 Asymptotic Expansions of Anger–Weber Functions
    16: 3.6 Linear Difference Equations
    3.6.9 | e N p N p N + 1 | ϵ min 1 n M | e n p n p n + 1 | .
    17: 24.16 Generalizations
    In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); p -adic integer order Bernoulli numbers (Adelberg (1996)); p -adic q -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli-Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).
    18: Bibliography M
  • 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.
  • 19: 14.15 Uniform Asymptotic Approximations
    §14.15(i) Large μ , Fixed ν
    For the interval - 1 < x < 1 with fixed ν , real μ , and arbitrary fixed values of the nonnegative integer J , … In this and subsequent subsections δ denotes an arbitrary constant such that 0 < δ < 1 . …
    14.15.5 α = ν + 1 2 μ ( < 1 ) ,
    For asymptotic expansions and explicit error bounds, see Dunster (2003b). …
    20: 14.1 Special Notation
    §14.1 Special Notation
    x , y , τ

    real variables.

    m , n

    unless stated otherwise, nonnegative integers, used for order and degree, respectively.

    μ , ν

    general order and degree, respectively.

    δ

    arbitrary small positive constant.