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11: 13.18 Relations to Other Functions
When 1 2 κ ± μ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). …
13.18.10 W 0 , 1 3 ( 4 3 z 3 2 ) = 2 π z 1 4 Ai ( z ) .
13.18.11 W 1 2 a , ± 1 4 ( 1 2 z 2 ) = 2 1 2 a z U ( a , z ) ,
13.18.12 M 1 2 a , 1 4 ( 1 2 z 2 ) = 2 1 2 a 1 Γ ( 1 2 a + 3 4 ) z / π ( U ( a , z ) + U ( a , z ) ) ,
13.18.13 M 1 2 a , 1 4 ( 1 2 z 2 ) = 2 1 2 a 2 Γ ( 1 2 a + 1 4 ) z / π ( U ( a , z ) U ( a , z ) ) .
12: 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 ) ,
10.67.10 ber x bei x ber x bei x e x 2 2 π x ( 1 2 + 1 8 1 x + 9 64 2 1 x 2 + 39 512 1 x 3 + 75 8192 2 1 x 4 + ) ,
10.67.11 ber x ber x + bei x bei x e x 2 2 π x ( 1 2 3 8 1 x 15 64 2 1 x 2 45 512 1 x 3 + 315 8192 2 1 x 4 + ) ,
10.67.14 ker x kei x ker x kei x π 2 x e x 2 ( 1 2 1 8 1 x + 9 64 2 1 x 2 39 512 1 x 3 + 75 8192 2 1 x 4 + ) ,
10.67.15 ker x ker x + kei x kei x π 2 x e x 2 ( 1 2 + 3 8 1 x 15 64 2 1 x 2 + 45 512 1 x 3 + 315 8192 2 1 x 4 + ) ,
13: 10.31 Power Series
10.31.1 K n ( z ) = 1 2 ( 1 2 z ) n k = 0 n 1 ( n k 1 ) ! k ! ( 1 4 z 2 ) k + ( 1 ) n + 1 ln ( 1 2 z ) I n ( z ) + ( 1 ) n 1 2 ( 1 2 z ) n k = 0 ( ψ ( k + 1 ) + ψ ( n + k + 1 ) ) ( 1 4 z 2 ) k k ! ( n + k ) ! ,
10.31.2 K 0 ( z ) = ( ln ( 1 2 z ) + γ ) I 0 ( z ) + 1 4 z 2 ( 1 ! ) 2 + ( 1 + 1 2 ) ( 1 4 z 2 ) 2 ( 2 ! ) 2 + ( 1 + 1 2 + 1 3 ) ( 1 4 z 2 ) 3 ( 3 ! ) 2 + .
10.31.3 I ν ( z ) I μ ( z ) = ( 1 2 z ) ν + μ k = 0 ( ν + μ + k + 1 ) k ( 1 4 z 2 ) k k ! Γ ( ν + k + 1 ) Γ ( μ + k + 1 ) .
14: 13.11 Series
13.11.1 M ( a , b , z ) = Γ ( a 1 2 ) e 1 2 z ( 1 4 z ) 1 2 a s = 0 ( 2 a 1 ) s ( 2 a b ) s ( b ) s s ! ( a 1 2 + s ) I a 1 2 + s ( 1 2 z ) , a + 1 2 , b 0 , 1 , 2 , ,
13.11.2 M ( a , b , z ) = Γ ( b a 1 2 ) e 1 2 z ( 1 4 z ) a b + 1 2 s = 0 ( 1 ) s ( 2 b 2 a 1 ) s ( b 2 a ) s ( b a 1 2 + s ) ( b ) s s ! I b a 1 2 + s ( 1 2 z ) , b a + 1 2 , b 0 , 1 , 2 , .
13.11.3 𝐌 ( a , b , z ) = e 1 2 z s = 0 A s ( b 2 a ) 1 2 ( 1 b s ) ( 1 2 z ) 1 2 ( 1 b + s ) J b 1 + s ( 2 z ( b 2 a ) ) ,
A 2 = 1 2 b ,
15: 12.5 Integral Representations
12.5.2 U ( a , z ) = z e 1 4 z 2 Γ ( 1 4 + 1 2 a ) 0 t 1 2 a 3 4 e t ( z 2 + 2 t ) 1 2 a 3 4 d t , | ph z | < 1 2 π , a > 1 2 ,
12.5.3 U ( a , z ) = e 1 4 z 2 Γ ( 3 4 + 1 2 a ) 0 t 1 2 a 1 4 e t ( z 2 + 2 t ) 1 2 a 1 4 d t , | ph z | < 1 2 π , a > 3 2 ,
12.5.4 U ( a , z ) = 2 π e 1 4 z 2 0 t a 1 2 e 1 2 t 2 cos ( z t + ( 1 2 a + 1 4 ) π ) d t , a < 1 2 .
where the contour separates the poles of Γ ( t ) from those of Γ ( 1 2 + a 2 t ) . … where the contour separates the poles of Γ ( t ) from those of Γ ( 1 2 a 2 t ) . …
16: 5.10 Continued Fractions
a 0 = 1 12 ,
a 1 = 1 30 ,
a 2 = 53 210 ,
a 3 = 195 371 ,
a 4 = 22999 22737 ,
17: 6.9 Continued Fraction
6.9.1 E 1 ( z ) = e z z + 1 1 + 1 z + 2 1 + 2 z + 3 1 + 3 z + , | ph z | < π .
18: 18.13 Continued Fractions
18.13.1 1 x + 1 2 x + 1 2 x + ,
18.13.2 1 2 x + 1 2 x + 1 2 x + .
18.13.3 a 1 x + 1 2 3 2 x + 2 3 5 3 x + 3 4 7 4 x + ,
18.13.4 a 1 1 x + 1 2 1 2 ( 3 x ) + 2 3 1 3 ( 5 x ) + 3 4 1 4 ( 7 x ) + ,
18.13.5 1 2 x + 2 2 x + 4 2 x + 6 2 x + .
19: 10.19 Asymptotic Expansions for Large Order
If ν through positive real values with β ( ( 0 , 1 2 π ) ) fixed, and … with sectors of validity 1 2 π + δ ± ph ν 3 2 π δ . …
P 4 ( a ) = 27 20000 a 10 23573 1 47000 a 7 + 5903 1 38600 a 4 + 947 3 46500 a ,
with sectors of validity 1 2 π + δ ph ν 3 2 π δ and 3 2 π + δ ph ν 1 2 π δ , respectively. …
R 4 ( a ) = 27 20000 a 10 46631 1 47000 a 7 + 3889 4620 a 4 1159 1 15500 a ,
20: 13.26 Addition and Multiplication Theorems
13.26.2 e 1 2 y ( x + y x ) μ + 1 2 n = 0 ( 1 2 + μ κ ) n ( 1 + 2 μ ) n n ! ( y x ) n M κ 1 2 n , μ + 1 2 n ( x ) ,
13.26.5 e 1 2 y ( x + y x ) μ + 1 2 n = 0 ( 1 2 + μ + κ ) n ( 1 + 2 μ ) n n ! ( y x ) n M κ + 1 2 n , μ + 1 2 n ( x ) ,
13.26.7 e 1 2 y ( x x + y ) μ 1 2 n = 0 ( 1 2 μ κ ) n n ! ( y x ) n W κ 1 2 n , μ 1 2 n ( x ) , | y | < | x | ,
13.26.10 e 1 2 y ( x x + y ) μ 1 2 n = 0 1 n ! ( y x ) n W κ + 1 2 n , μ 1 2 n ( x ) , | y | < | x | ,
13.26.11 e 1 2 y ( x + y x ) μ + 1 2 n = 0 1 n ! ( y x ) n W κ + 1 2 n , μ + 1 2 n ( x ) , | y | < | x | ,