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31: 28.22 Connection Formulas
28.22.5 g e , 2 m ( h ) = ( 1 ) m 2 π ce 2 m ( 1 2 π , h 2 ) A 0 2 m ( h 2 ) ,
28.22.6 g e , 2 m + 1 ( h ) = ( 1 ) m + 1 2 π ce 2 m + 1 ( 1 2 π , h 2 ) h A 1 2 m + 1 ( h 2 ) ,
28.22.7 g o , 2 m + 1 ( h ) = ( 1 ) m 2 π se 2 m + 1 ( 1 2 π , h 2 ) h B 1 2 m + 1 ( h 2 ) ,
32: 28.29 Definitions and Basic Properties
28.29.2 Q ( z + π ) = Q ( z ) ,
28.29.11 w ( z + π ) = ( 1 ) ν w ( z ) + c P ( z ) ,
28.29.13 w ( z + π ) + w ( z π ) = 2 cos ( π ν ) w ( z ) .
33: 7.11 Relations to Other Functions
7.11.4 erf z = 2 z π M ( 1 2 , 3 2 , z 2 ) = 2 z π e z 2 M ( 1 , 3 2 , z 2 ) ,
7.11.5 erfc z = 1 π e z 2 U ( 1 2 , 1 2 , z 2 ) = z π e z 2 U ( 1 , 3 2 , z 2 ) .
34: 9.5 Integral Representations
9.5.1 Ai ( x ) = 1 π 0 cos ( 1 3 t 3 + x t ) d t .
9.5.2 Ai ( x ) = x 1 / 2 π 1 cos ( x 3 / 2 ( 1 3 t 3 + t 2 2 3 ) ) d t , x > 0 .
9.5.4 Ai ( z ) = 1 2 π i e π i / 3 e π i / 3 exp ( 1 3 t 3 z t ) d t ,
9.5.6 Ai ( z ) = 3 2 π 0 exp ( t 3 3 z 3 3 t 3 ) d t , | ph z | < 1 6 π .
9.5.7 Ai ( z ) = e ζ π 0 exp ( z 1 / 2 t 2 ) cos ( 1 3 t 3 ) d t , | ph z | < π .
35: 24.8 Series Expansions
24.8.1 B 2 n ( x ) = ( 1 ) n + 1 2 ( 2 n ) ! ( 2 π ) 2 n k = 1 cos ( 2 π k x ) k 2 n ,
24.8.2 B 2 n + 1 ( x ) = ( 1 ) n + 1 2 ( 2 n + 1 ) ! ( 2 π ) 2 n + 1 k = 1 sin ( 2 π k x ) k 2 n + 1 .
24.8.3 B n ( x ) = n ! ( 2 π i ) n k = k 0 e 2 π i k x k n .
24.8.6 B 4 n + 2 = ( 8 n + 4 ) k = 1 k 4 n + 1 e 2 π k 1 , n = 1 , 2 , ,
24.8.7 B 2 n = ( 1 ) n + 1 4 n 2 2 n 1 k = 1 k 2 n 1 e π k + ( 1 ) k + n , n = 2 , 3 , .
36: 28.19 Expansions in Series of me ν + 2 n Functions
28.19.3 f n = 1 π 0 π f ( z ) me ν + 2 n ( z , q ) d z .
37: 28.30 Expansions in Series of Eigenfunctions
28.30.4 f m = 1 2 π 0 2 π f ( x ) w m ( x ) d x .
38: 5.16 Sums
5.16.1 k = 1 ( 1 ) k ψ ( k ) = π 2 8 ,
39: 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 ) ,
10.67.2 kei ν x e x / 2 ( π 2 x ) 1 2 k = 0 a k ( ν ) x k sin ( x 2 + ( ν 2 + k 4 + 1 8 ) π ) .
10.67.5 ker ν x e x / 2 ( π 2 x ) 1 2 k = 0 b k ( ν ) x k cos ( x 2 + ( ν 2 + k 4 1 8 ) π ) ,
10.67.6 kei ν x e x / 2 ( π 2 x ) 1 2 k = 0 b k ( ν ) x k sin ( x 2 + ( ν 2 + k 4 1 8 ) π ) .
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 + ) ,
40: 33.18 Limiting Forms for Large