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11: Bibliography C
  • J. B. Campbell (1979) Bessel functions J ν ( x ) and Y ν ( x ) of real order and real argument. Comput. Phys. Comm. 18 (1), pp. 133–142.
  • J. B. Campbell (1981) Bessel functions I ν ( x ) and K ν ( x ) of real order and complex argument. Comput. Phys. Comm. 24 (1), pp. 97–105.
  • R. M. Corless, D. J. Jeffrey, and H. Rasmussen (1992) Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.
  • 12: 19.7 Connection Formulas
    19.7.8 Π ( ϕ , α 2 , k ) + Π ( ϕ , ω 2 , k ) = F ( ϕ , k ) + c R C ( ( c 1 ) ( c k 2 ) , ( c α 2 ) ( c ω 2 ) ) , α 2 ω 2 = k 2 .
    19.7.9 ( k 2 α 2 ) Π ( ϕ , α 2 , k ) + ( k 2 ω 2 ) Π ( ϕ , ω 2 , k ) = k 2 F ( ϕ , k ) α 2 ω 2 c 1 R C ( c ( c k 2 ) , ( c α 2 ) ( c ω 2 ) ) , ( 1 α 2 ) ( 1 ω 2 ) = 1 k 2 .
    19.7.10 ( 1 α 2 ) Π ( ϕ , α 2 , k ) + ( 1 ω 2 ) Π ( ϕ , ω 2 , k ) = F ( ϕ , k ) + ( 1 α 2 ω 2 ) c k 2 R C ( c ( c 1 ) , ( c α 2 ) ( c ω 2 ) ) , ( k 2 α 2 ) ( k 2 ω 2 ) = k 2 ( k 2 1 ) .
    13: 15.19 Methods of Computation
    For z it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval [ 0 , 1 2 ] . …
    14: 14.20 Conical (or Mehler) Functions
    For extensions to complex arguments (including the range 1 < x < ), asymptotic expansions, and explicit error bounds, see Dunster (1991). …
    15: Bibliography O
  • C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.
  • 16: Errata
  • Section 1.13

    In Equation (1.13.4), the determinant form of the two-argument Wronskian

    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) w 2 ( z ) w 1 ( z )

    was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

  • Equation (35.7.3)

    Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument F 1 2 was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

  • Equation (18.27.6)

    18.27.6 P n ( α , β ) ( x ; c , d ; q ) = c n q ( α + 1 ) n ( q α + 1 , q α + 1 c 1 d ; q ) n ( q , q ; q ) n P n ( q α + 1 c 1 x ; q α , q β , q α c 1 d ; q )

    Originally the first argument to the big q -Jacobi polynomial on the right-hand side was written incorrectly as q α + 1 c 1 d x .

    Reported 2017-09-27 by Tom Koornwinder.

  • Equation (10.17.14)
    10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t ) exp ( | ν 2 1 4 | 𝒱 z , ± i ( t 1 ) )

    Originally the factor 𝒱 z , ± i ( t 1 ) in the argument to the exponential was written incorrectly as 𝒱 z , ± i ( t ) .

    Reported 2014-09-27 by Gergő Nemes.

  • Equation (22.19.2)
    22.19.2 sin ( 1 2 θ ( t ) ) = sin ( 1 2 α ) sn ( t + K , sin ( 1 2 α ) )

    Originally the first argument to the function sn was given incorrectly as t . The correct argument is t + K .

    Reported 2014-03-05 by Svante Janson.

  • 17: 16.4 Argument Unity
    The function F q q + 1 with argument unity and general values of the parameters is discussed in Bühring (1992). …
    18: 17.11 Transformations of q -Appell Functions
    17.11.1 Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) = ( a , b x , b y ; q ) ( c , x , y ; q ) ϕ 2 3 ( c / a , x , y b x , b y ; q , a ) ,
    17.11.2 Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) = ( b , a x ; q ) ( c , x ; q ) n , r 0 ( a , b ; q ) n ( c / b , x ; q ) r b r y n ( q , c ; q ) n ( q ; q ) r ( a x ; q ) n + r ,
    17.11.3 Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) = ( a , b x ; q ) ( c , x ; q ) n , r 0 ( a , b ; q ) n ( x ; q ) r ( c / a ; q ) n + r a r y n ( q , c / a ; q ) n ( q , b x ; q ) r .
    19: Bibliography S
  • J. Segura and A. Gil (1998) Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.
  • 20: 4.15 Graphics
    §4.15(i) Real Arguments
    §4.15(ii) Complex Arguments: Conformal Maps
    Figure 4.15.7 illustrates the conformal mapping of the strip 1 2 π < z < 1 2 π onto the whole w -plane cut along the real axis from to 1 and 1 to , where w = sin z and z = arcsin w (principal value). …
    §4.15(iii) Complex Arguments: Surfaces
    The corresponding surfaces for arccos ( x + i y ) , arccot ( x + i y ) , arcsec ( x + i y ) can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).