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11: Bibliography D
  • T. M. Dunster (1986) Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.
  • T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
  • T. M. Dunster (1991) Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
  • T. M. Dunster (1992) Uniform asymptotic expansions for oblate spheroidal functions I: Positive separation parameter λ . Proc. Roy. Soc. Edinburgh Sect. A 121 (3-4), pp. 303–320.
  • T. M. Dunster (1995) Uniform asymptotic expansions for oblate spheroidal functions II: Negative separation parameter λ . Proc. Roy. Soc. Edinburgh Sect. A 125 (4), pp. 719–737.
  • 12: 36.10 Differential Equations
    36.10.5 4 Ψ 3 x 4 3 5 z 2 Ψ 3 x 2 2 i 5 y Ψ 3 x + 1 5 x Ψ 3 = 0 .
    36.10.13 6 2 Ψ ( E ) x y 2 i z Ψ ( E ) y + y Ψ ( E ) = 0 ,
    13: 15.11 Riemann’s Differential Equation
    15.11.2 a 1 + a 2 + b 1 + b 2 + c 1 + c 2 = 1 .
    Here { a 1 , a 2 } , { b 1 , b 2 } , { c 1 , c 2 } are the exponent pairs at the points α , β , γ , respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice { 0 , 1 } for exponent pairs. …
    15.11.3 w = P { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } .
    14: 15.10 Hypergeometric Differential Equation
    It has regular singularities at z = 0 , 1 , , with corresponding exponent pairs { 0 , 1 c } , { 0 , c a b } , { a , b } , respectively. When none of the exponent pairs differ by an integer, that is, when none of c , c a b , a b is an integer, we have the following pairs f 1 ( z ) , f 2 ( z ) of fundamental solutions. …
    15.10.3 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( 1 c ) z c ( 1 z ) c a b 1 .
    15.10.5 𝒲 { f 1 ( z ) , f 2 ( z ) } = ( a + b c ) z c ( 1 z ) c a b 1 .
    (c) If the parameter c in the differential equation equals 2 n = 0 , 1 , 2 , , then fundamental solutions in the neighborhood of z = 0 are given by z n 1 times those in (a) and (b), with a and b replaced throughout by a + n 1 and b + n 1 , respectively. …
    15: 36.5 Stokes Sets
    36.5.2 y 3 = 27 4 ( 27 5 ) x 2 = 1.32403 x 2 .
    36.5.4 80 x 5 40 x 4 55 x 3 + 5 x 2 + 20 x 1 = 0 ,
    36.5.11 x z 2 = 1 12 u 2 + 8 u | y z 2 | 1 3 u ( u ( 2 3 u ) ) 1 / 2 .
    36.5.12 8 u 3 4 u 2 | y 3 z 2 | ( u 2 3 u ) 1 / 2 = y 2 6 w z 4 2 w 3 2 w 2 ,
    36.5.13 w = u 2 3 + ( ( 2 3 u ) 2 + | y 6 z 2 | ( 2 3 u u ) 1 / 2 ) 1 / 2 ,
    16: 36.6 Scaling Relations
    Table 36.6.1: Special cases of scaling exponents for cuspoids.
    singularity K β K γ 1 K γ 2 K γ 3 K γ K
    For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).
    17: 2.5 Mellin Transform Methods
    where J ν denotes the Bessel function (§10.2(ii)), and x is a large positive parameter. … Then as in (2.5.6) and (2.5.7), with d = 2 n + 1 ϵ   ( 0 < ϵ < 1 ) , we obtain …
    §2.5(iii) Laplace Transforms with Small Parameters
    If κ = 0 in (2.5.18) and c > 1 in (2.5.20), and if none of the exponents in (2.5.18) are positive integers, then the expansion (2.5.43) gives the following useful result: … For examples in which the integral defining the Mellin transform h ( z ) does not exist for any value of z , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
    18: 36.8 Convergent Series Expansions
    36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 𝐱 ) = Ai ( x ) Ai ( y ) n = 0 ( 3 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( 3 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( 3 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( 3 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
    36.8.4 Ψ ( E ) ( 𝐱 ) = 2 π 2 ( 2 3 ) 2 / 3 n = 0 ( i ( 2 / 3 ) 2 / 3 z ) n n ! ( f n ( x + i y 12 1 / 3 , x i y 12 1 / 3 ) ) ,
    19: 28.34 Methods of Computation
    §28.34(i) Characteristic Exponents
  • (b)

    Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

  • 20: Bibliography M
  • E. L. Mansfield and H. N. Webster (1998) On one-parameter families of Painlevé III. Stud. Appl. Math. 101 (3), pp. 321–341.
  • N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.
  • mpmath (free python library)
  • H. P. Mulholland and S. Goldstein (1929) The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.
  • J. Murzewski and A. Sowa (1972) Tables of the functions of the parabolic cylinder for negative integer parameters. Zastos. Mat. 13, pp. 261–273.