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1: 2.8 Differential Equations with a Parameter
of smallest absolute value, and define the envelopes of Ai ( x ) and Bi ( x ) by …These envelopes are continuous functions of x , and as u Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. …Define … (For envelope functions for parabolic cylinder functions see §14.15(v)). …
2: 13.21 Uniform Asymptotic Approximations for Large κ
13.21.1 M κ , μ ( x ) = x Γ ( 2 μ + 1 ) κ - μ ( J 2 μ ( 2 x κ ) + env J 2 μ ( 2 x κ ) O ( κ - 1 2 ) ) ,
13.21.2 W κ , μ ( x ) = x Γ ( κ + 1 2 ) ( sin ( κ π - μ π ) J 2 μ ( 2 x κ ) - cos ( κ π - μ π ) Y 2 μ ( 2 x κ ) + env Y 2 μ ( 2 x κ ) O ( κ - 1 2 ) ) ,
13.21.13 M κ , μ ( x ) = Γ ( 2 μ + 1 ) ( e 2 κ 2 - μ 2 ) 1 2 μ ( κ - μ κ + μ ) 1 2 κ Ψ ( κ , μ , x ) ( J 2 μ ( ζ κ ) + env J 2 μ ( ζ κ ) O ( κ - 1 ) ) ,
13.21.14 W κ , μ ( x ) = e - μ π i π Γ ( κ + μ + 1 2 ) Γ ( κ - μ + 1 2 ) c ( κ , μ ) Ψ ( κ , μ , x ) ( sin ( κ π - μ π ) J 2 μ ( ζ κ ) - cos ( κ π - μ π ) Y 2 μ ( ζ κ ) + env Y 2 μ ( ζ κ ) O ( κ - 1 ) ) ,
13.21.22 M κ , μ ( x ) = 1 2 π Γ ( 2 μ + 1 ) Γ ( κ - μ + 1 2 ) c ^ ( κ , μ ) Ψ ^ ( κ , μ , x ) ( sin ( κ π - μ π ) Ai ( κ 2 3 ζ ^ ) + cos ( κ π - μ π ) Bi ( κ 2 3 ζ ^ ) + envBi ( κ 2 3 ζ ^ ) O ( κ - 1 ) ) ,
3: 33.3 Graphics
33.3.1 M ( η , ρ ) = ( F 2 ( η , ρ ) + G 2 ( η , ρ ) ) 1 / 2 = | H ± ( η , ρ ) | .
4: 13.20 Uniform Asymptotic Approximations for Large μ
13.20.16 W κ , μ ( x ) = ( 1 2 μ ) - 1 4 ( κ + μ e ) 1 2 ( κ + μ ) Φ ( κ , μ , x ) ( U ( μ - κ , ζ 2 μ ) + env U ( μ - κ , ζ 2 μ ) O ( μ - 2 3 ) ) ,
13.20.17 M κ , μ ( x ) = ( 8 μ ) 1 4 ( 2 μ e ) 2 μ ( e κ + μ ) 1 2 ( κ + μ ) Φ ( κ , μ , x ) ( U ( μ - κ , - ζ 2 μ ) + env U ¯ ( μ - κ , ζ 2 μ ) O ( μ - 2 3 ) ) ,
13.20.18 W κ , μ ( x ) = ( 1 2 μ ) - 1 4 ( κ + μ e ) 1 2 ( κ + μ ) Φ ( κ , μ , x ) ( U ( μ - κ , ζ 2 μ ) + env U ¯ ( μ - κ , - ζ 2 μ ) O ( μ - 2 3 ) ) ,
13.20.19 M κ , μ ( x ) = ( 8 μ ) 1 4 ( 2 μ e ) 2 μ ( e κ + μ ) 1 2 ( κ + μ ) Φ ( κ , μ , x ) ( U ( μ - κ , - ζ 2 μ ) + env U ( μ - κ , - ζ 2 μ ) O ( μ - 2 3 ) ) ,
5: 14.15 Uniform Asymptotic Approximations
14.15.11 P ν - μ ( cos θ ) = 1 ν μ ( θ sin θ ) 1 / 2 ( J μ ( ( ν + 1 2 ) θ ) + O ( 1 ν ) env J μ ( ( ν + 1 2 ) θ ) ) ,
14.15.15 P ν - μ ( x ) = β ( y - α 2 1 - α 2 - x 2 ) 1 / 4 ( J μ ( ( ν + 1 2 ) y 1 / 2 ) + O ( 1 ν ) env J μ ( ( ν + 1 2 ) y 1 / 2 ) ) ,
Here we introduce the envelopes of the parabolic cylinder functions U ( - c , x ) , U ¯ ( - c , x ) , which are defined in §12.2. For U ( - c , x ) or U ¯ ( - c , x ) , with c and x nonnegative, …
6: 13.8 Asymptotic Approximations for Large Parameters
13.8.9 M ( a , b , x ) = Γ ( b ) e 1 2 x ( ( 1 2 b - a ) x ) 1 2 - 1 2 b ( J b - 1 ( 2 x ( b - 2 a ) ) + env J b - 1 ( 2 x ( b - 2 a ) ) O ( | a | - 1 2 ) ) ,
13.8.10 U ( a , b , x ) = Γ ( 1 2 b - a + 1 2 ) e 1 2 x x 1 2 - 1 2 b ( cos ( a π ) J b - 1 ( 2 x ( b - 2 a ) ) - sin ( a π ) Y b - 1 ( 2 x ( b - 2 a ) ) + env Y b - 1 ( 2 x ( b - 2 a ) ) O ( | a | - 1 2 ) ) ,
7: 18.15 Asymptotic Approximations
18.15.19 L n ( α ) ( ν x ) = e 1 2 ν x 2 α x 1 2 α + 1 4 ( 1 - x ) 1 4 ( ξ 1 2 J α ( ν ξ ) m = 0 M - 1 A m ( ξ ) ν 2 m + ξ - 1 2 J α + 1 ( ν ξ ) m = 0 M - 1 B m ( ξ ) ν 2 m + 1 + ξ 1 2 env J α ( ν ξ ) O ( 1 ν 2 M - 1 ) ) ,
Here J ν ( z ) denotes the Bessel function10.2(ii)), env J ν ( z ) denotes its envelope2.8(iv)), and δ is again an arbitrary small positive constant. …
18.15.22 L n ( α ) ( ν x ) = ( - 1 ) n e 1 2 ν x 2 α - 1 2 x 1 2 α + 1 4 ( ζ x - 1 ) 1 4 ( Ai ( ν 2 3 ζ ) ν 1 3 m = 0 M - 1 E m ( ζ ) ν 2 m + Ai ( ν 2 3 ζ ) ν 5 3 m = 0 M - 1 F m ( ζ ) ν 2 m + envAi ( ν 2 3 ζ ) O ( 1 ν 2 M - 2 3 ) ) ,
Here Ai denotes the Airy function9.2), Ai denotes its derivative, and envAi denotes its envelope2.8(iii)). …
8: Errata
  • Equation (14.15.23)

    Originally used f ( x ) to represent both U ( - c , x ) and U ¯ ( - c , x ) . This has been replaced by two equations giving explicit definitions for the two envelope functions. Some slight changes in wording were needed to make this clear to readers.

  • 9: 12.2 Differential Equations
    §12.2 Differential Equations
    §12.2(i) Introduction
    All solutions are entire functions of z and entire functions of a or ν . …
    §12.2(iii) Wronskians
    Its importance is that when a is negative and | a | is large, U ( a , x ) and U ¯ ( a , x ) asymptotically have the same envelope (modulus) and are 1 2 π out of phase in the oscillatory interval - 2 - a < x < 2 - a . …