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1: Bibliography Z
  • M. I. Žurina and L. N. Karmazina (1963) Tablitsy funktsii Lezhandra P - 1 / 2 + i τ 1 ( x ) . Vyčisl. Centr Akad. Nauk SSSR, Moscow.
  • M. I. Žurina and L. N. Karmazina (1964) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.
  • M. I. Žurina and L. N. Karmazina (1965) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.
  • M. I. Žurina and L. N. Karmazina (1966) Tables and formulae for the spherical functions P - 1 / 2 + i τ m ( z ) . Translated by E. L. Albasiny, Pergamon Press, Oxford.
  • 2: 14.31 Other Applications
    The conical functions P - 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
    3: 14.33 Tables
  • Žurina and Karmazina (1964, 1965) tabulate the conical functions P - 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = - 0.9 ( .1 ) 0.9 , 7S; P - 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7D. Auxiliary tables are included to facilitate computation for larger values of τ when - 1 < x < 1 .

  • Žurina and Karmazina (1963) tabulates the conical functions P - 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = - 0.9 ( .1 ) 0.9 , 7S; P - 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7S. Auxiliary tables are included to assist computation for larger values of τ when - 1 < x < 1 .

  • 4: 14.1 Special Notation
    The main functions treated in this chapter are the Legendre functions P ν ( x ) , Q ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions P ν μ ( x ) , Q ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , Q ν μ ( z ) ; conical functions P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) , Q ^ - 1 2 + i τ μ ( x ) , P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) (also known as Mehler functions). …
    5: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2009) Computing the conical function P - 1 / 2 + i τ μ ( x ) . SIAM J. Sci. Comput. 31 (3), pp. 1716–1741.
  • A. Gil, J. Segura, and N. M. Temme (2012) An improved algorithm and a Fortran 90 module for computing the conical function P - 1 / 2 + i τ m ( x ) . Comput. Phys. Commun. 183 (3), pp. 794–799.
  • 6: 14.34 Software
    A more complete list of available software for computing these functions is found in the Software Index. For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C9).
    §14.34(ii) Legendre Functions: Real Argument and Parameters
    §14.34(iii) Legendre Functions: Complex Argument and/or Parameters
    §14.34(iv) Conical (Mehler) and/or Toroidal Functions
    7: 14.27 Zeros
    §14.27 Zeros
    P ν μ ( x ± i 0 ) (either side of the cut) has exactly one zero in the interval ( - , - 1 ) if either of the following sets of conditions holds:
  • (a)

    μ < 0 , μ , ν , and sin ( ( μ - ν ) π ) and sin ( μ π ) have opposite signs.

  • For all other values of the parameters P ν μ ( x ± i 0 ) has no zeros in the interval ( - , - 1 ) . For complex zeros of P ν μ ( z ) see Hobson (1931, §§233, 234, and 238).
    8: 14.20 Conical (or Mehler) Functions
    For - 1 < x < 1 and τ > 0 , a numerically satisfactory pair of real conical functions is P - 1 2 + i τ - μ ( x ) and P - 1 2 + i τ - μ ( - x ) . …
    14.20.2 Q ^ - 1 2 + i τ - μ ( x ) = ( e μ π i Q - 1 2 + i τ - μ ( x ) ) - 1 2 π sin ( μ π ) P - 1 2 + i τ - μ ( x ) .
    9: 23.2 Definitions and Periodic Properties
    23.2.4 ( z ) = 1 z 2 + w 𝕃 { 0 } ( 1 ( z - w ) 2 - 1 w 2 ) ,
    10: 7.1 Special Notation
    Alternative notations are Q ( z ) = 1 2 erfc ( z / 2 ) , P ( z ) = Φ ( z ) = 1 2 erfc ( - z / 2 ) , Erf z = 1 2 π erf z , Erfi z = e z 2 F ( z ) , C 1 ( z ) = C ( 2 / π z ) , S 1 ( z ) = S ( 2 / π z ) , C 2 ( z ) = C ( 2 z / π ) , S 2 ( z ) = S ( 2 z / π ) . …