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1: 16.22 Asymptotic Expansions
§16.22 Asymptotic Expansions
Asymptotic expansions of G p , q m , n ( z ; a ; b ) for large z are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer G -functions with large parameters see Fields (1973, 1983).
2: 10.70 Zeros
zeros of  ber ν x 2 ( t - f ( t ) ) , t = ( m - 1 2 ν - 3 8 ) π ,
zeros of  bei ν x 2 ( t - f ( t ) ) , t = ( m - 1 2 ν + 1 8 ) π ,
zeros of  ker ν x 2 ( t + f ( - t ) ) , t = ( m - 1 2 ν - 5 8 ) π ,
zeros of  kei ν x 2 ( t + f ( - t ) ) , t = ( m - 1 2 ν - 1 8 ) π .
3: 10.67 Asymptotic Expansions for Large Argument
§10.67 Asymptotic Expansions for Large Argument
10.67.1 ker ν x e - x / 2 ( π 2 x ) 1 2 k = 0 a k ( ν ) x k cos ( x 2 + ( ν 2 + k 4 + 1 8 ) π ) ,
§10.67(ii) Cross-Products and Sums of Squares in the Case ν = 0
10.67.9 ber 2 x + bei 2 x e x 2 2 π x ( 1 + 1 4 2 1 x + 1 64 1 x 2 - 33 256 2 1 x 3 - 1797 8192 1 x 4 + ) ,
4: 28.16 Asymptotic Expansions for Large q
§28.16 Asymptotic Expansions for Large q
28.16.1 λ ν ( h 2 ) - 2 h 2 + 2 s h - 1 8 ( s 2 + 1 ) - 1 2 7 h ( s 3 + 3 s ) - 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) - 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) - 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) - 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
5: 10.69 Uniform Asymptotic Expansions for Large Order
§10.69 Uniform Asymptotic Expansions for Large Order
10.69.2 ber ν ( ν x ) + i bei ν ( ν x ) e ν ξ ( 2 π ν ξ ) 1 / 2 ( x e 3 π i / 4 1 + ξ ) ν k = 0 U k ( ξ - 1 ) ν k ,
All fractional powers take their principal values. …
6: 12.9 Asymptotic Expansions for Large Variable
§12.9 Asymptotic Expansions for Large Variable
12.9.1 U ( a , z ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , | ph z | 3 4 π - δ ( < 3 4 π ) ,
12.9.2 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s , | ph z | 1 4 π - δ ( < 1 4 π ) .
12.9.4 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s ± i Γ ( 1 2 - a ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , - 1 4 π + δ ± ph z 3 4 π - δ .
§12.9(ii) Bounds and Re-Expansions for the Remainder Terms
7: 6.12 Asymptotic Expansions
§6.12 Asymptotic Expansions
§6.12(i) Exponential and Logarithmic Integrals
For the function χ see §9.7(i). …
§6.12(ii) Sine and Cosine Integrals
8: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
9: 13.19 Asymptotic Expansions for Large Argument
§13.19 Asymptotic Expansions for Large Argument
13.19.1 M κ , μ ( x ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) e 1 2 x x - κ s = 0 ( 1 2 - μ + κ ) s ( 1 2 + μ + κ ) s s ! x - s , μ - κ - 1 2 , - 3 2 , .
For an asymptotic expansion of W κ , μ ( z ) as z that is valid in the sector | ph z | π - δ and where the real parameters κ , μ are subject to the growth conditions κ = o ( z ) , μ = o ( z ) , see Wong (1973a).
10: 2.2 Transcendental Equations
2.2.6 t = y 1 2 ( 1 + 1 4 y - 1 ln y + o ( y - 1 ) ) , y .
An important case is the reversion of asymptotic expansions for zeros of special functions. …
2.2.7 f ( x ) x + f 0 + f 1 x - 1 + f 2 x - 2 + , x .
2.2.8 x y - F 0 - F 1 y - 1 - F 2 y - 2 - , y ,
where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …