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1: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
§9.13(i) Generalizations from the Differential Equation
and 𝒵 p is any linear combination of the modified Bessel functions I p and e p π i K p 10.25(ii)). … As z
2: Bibliography J
  • S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions p s ¯ n r ( η , h ) and q s ¯ n r ( η , h ) for large h . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.
  • 3: 9.18 Tables
  • Smirnov (1960) tabulates U 1 ( x , α ) , U 2 ( x , α ) , defined by (9.13.20), (9.13.21), and also U 1 ( x , α ) / x , U 2 ( x , α ) / x , for α = 1 , x = 6 ( .01 ) 10 to 5D or 5S, and also for α = ± 1 4 , ± 1 3 , ± 1 2 , ± 2 3 , ± 3 4 , 5 4 , 4 3 , 3 2 , 5 3 , 7 4 , 2, x = 0 ( .01 ) 6 ; 4D.

  • 4: 22.10 Maclaurin Series
    Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …
    5: 2.7 Differential Equations
    The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from z 0 . …
    6: 10.36 Other Differential Equations
    Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing z by i z .
    7: 19.18 Derivatives and Differential Equations
    §19.18 Derivatives and Differential Equations
    §19.18(ii) Differential Equations
    and also a system of n ( n 1 ) / 2 Euler–Poisson differential equations (of which only n 1 are independent): …If n = 2 , then elimination of 2 v between (19.18.11) and (19.18.12), followed by the substitution ( b 1 , b 2 , z 1 , z 2 ) = ( b , c b , 1 z , 1 ) , produces the Gauss hypergeometric equation (15.10.1). The next four differential equations apply to the complete case of R F and R G in the form R a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). …
    8: 10.13 Other Differential Equations
    §10.13 Other Differential Equations
    In the following equations ν , λ , p , q , and r are real or complex constants with λ 0 , p 0 , and q 0 .
    10.13.1 w ′′ + ( λ 2 ν 2 1 4 z 2 ) w = 0 , w = z 1 2 𝒞 ν ( λ z ) ,
    10.13.4 w ′′ + 1 2 ν z w + λ 2 w = 0 , w = z ± ν 𝒞 ν ( λ z ) ,
    For further differential equations see Kamke (1977, pp. 440–451). …
    9: 3.11 Approximation Techniques
    3.11.26 F ( s ) = f ( s ) = 0 e s t f ( t ) d t
    From the equations S / a k = 0 , k = 0 , 1 , , n , we derive the normal equations
    10: Bibliography G
  • J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.