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21: 28.12 Definitions and Basic Properties
For change of signs of ν and q , … For changes of sign of ν , q , and z ,
28.12.8 me - ν ( z , q ) = me ν ( - z , q ) ,
For change of signs of ν and z ,
28.12.14 ce ν ( z , q ) = ce ν ( - z , q ) = ce - ν ( z , q ) ,
22: 2.5 Mellin Transform Methods
2.5.41 I 1 ( x ) = h 1 ( 1 ) x - 1 + 1 2 π i ρ - i ρ + i x - z Γ ( 1 - z ) h 1 ( z ) d z ,
2.5.42 I 2 ( x ) = β 0 z 1 res [ - x - z Γ ( 1 - z ) h 2 ( z ) ] + 1 2 π i ρ - i ρ + i x - z Γ ( 1 - z ) h 2 ( z ) d z .
23: 1.14 Integral Transforms
1.14.33 lim t ± f ( σ + i t ) = 0 .
1.14.34 1 2 ( f ( u + ) + f ( u - ) ) = 1 2 π i lim T σ - i T σ + i T u - s f ( s ) d s .
1.14.35 f ( x ) = 1 2 π i σ - i σ + i x - s f ( s ) d s .
1.14.36 0 f ( x ) g ( y x ) d x = 1 2 π i σ - i σ + i y - s f ( 1 - s ) g ( s ) d s ,
1.14.49 lim t 0 + 𝒮 f ( - σ - i t ) - 𝒮 f ( - σ + i t ) 2 π i = 1 2 ( f ( σ + ) + f ( σ - ) ) ,
24: 19.2 Definitions
19.2.11 cel ( k c , p , a , b ) = 0 π / 2 a cos 2 θ + b sin 2 θ cos 2 θ + p sin 2 θ d θ cos 2 θ + k c 2 sin 2 θ ,
Here a , b , p are real parameters, and k c and x are real or complex variables, with p 0 , k c 0 . …
k c = k ,
p = 1 - α 2 ,
x = tan ϕ ,
25: 11.11 Asymptotic Expansions of Anger–Weber Functions
For fixed λ ( > 0 ) , …
11.11.9_5 a k + 1 ( λ ) = λ 1 - λ 2 λ a k ′′ ( λ ) + a k ( λ ) ( 2 k + 1 ) ( 2 k + 2 ) , k = 0 , 1 , 2 , .
11.11.12 μ = 1 - λ 2 - ln ( 1 + 1 - λ 2 λ ) ,
11.11.13_5 ( 1 2 ) k b k ( λ ) = ( - 1 ) k ( 1 - λ 2 ) 1 / 4 U k ( 1 1 - λ 2 ) , k = 0 , 1 , 2 , ,
11.11.14 A - ν ( λ ν ) 1 π ν ( λ - 1 ) , λ > 1 , | ph ν | π - δ ,
26: 12.11 Zeros
§12.11(iii) Asymptotic Expansions for Large Parameter
12.11.4 u a , s 2 1 2 μ ( p 0 ( α ) + p 1 ( α ) μ 4 + p 2 ( α ) μ 8 + ) ,
12.11.5 p 0 ( ζ ) = t ( ζ ) ,
12.11.6 p 1 ( ζ ) = t 3 - 6 t 24 ( t 2 - 1 ) 2 + 5 48 ( ( t 2 - 1 ) ζ 3 ) 1 2 .
12.11.8 q 0 ( ζ ) = t ( ζ ) .
27: 28.4 Fourier Series
28.4.10 m = 0 ( A 2 m + 1 2 n + 1 ( q ) ) 2 = 1 ,
28.4.11 m = 0 ( B 2 m + 1 2 n + 1 ( q ) ) 2 = 1 ,
28.4.12 m = 0 ( B 2 m + 2 2 n + 2 ( q ) ) 2 = 1 .
§28.4(v) Change of Sign of q
28.4.17 A 2 m 2 n ( - q ) = ( - 1 ) n - m A 2 m 2 n ( q ) ,
28: 12.14 The Function W ( a , x )
These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument z and parameter a . …
§12.14(ix) Uniform Asymptotic Expansions for Large Parameter
Positive a , 2 a < x <
Airy-type Uniform Expansions
29: 25.11 Hurwitz Zeta Function
25.11.15 ζ ( s , k a ) = k - s n = 0 k - 1 ζ ( s , a + n k ) , s 1 , k = 1 , 2 , 3 , .
25.11.31 1 Γ ( s ) 0 x s - 1 e - a x 2 cosh x d x = 4 - s ( ζ ( s , 1 4 + 1 4 a ) - ζ ( s , 3 4 + 1 4 a ) ) , s > 0 , a > - 1 .
25.11.35 n = 0 ( - 1 ) n ( n + a ) s = 1 Γ ( s ) 0 x s - 1 e - a x 1 + e - x d x = 2 - s ( ζ ( s , 1 2 a ) - ζ ( s , 1 2 ( 1 + a ) ) ) , a > 0 , s > 0 ; or a = 0 , a 0 , 0 < s < 1 .
30: 20.11 Generalizations and Analogs
20.11.4 f ( a , b ) = θ 3 ( z | τ ) .
§20.11(iii) Ramanujan’s Change of Base
These results are called Ramanujan’s changes of base. …