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1: 8.18 Asymptotic Expansions of I x ( a , b )
General Case
Let Γ ~ ( z ) denote the scaled gamma function
8.18.13 Γ ~ ( z ) = ( 2 π ) - 1 / 2 e z z ( 1 / 2 ) - z Γ ( z ) ,
2: 14.3 Definitions and Hypergeometric Representations
14.3.4 P ν m ( x ) = ( - 1 ) m Γ ( ν + m + 1 ) 2 m Γ ( ν - m + 1 ) ( 1 - x 2 ) m / 2 F ( ν + m + 1 , m - ν ; m + 1 ; 1 2 - 1 2 x ) ;
14.3.5 P ν m ( x ) = ( - 1 ) m Γ ( ν + m + 1 ) Γ ( ν - m + 1 ) ( 1 - x 1 + x ) m / 2 F ( ν + 1 , - ν ; m + 1 ; 1 2 - 1 2 x ) .
14.3.13 w 1 ( ν , μ , x ) = 2 μ Γ ( 1 2 ν + 1 2 μ + 1 2 ) Γ ( 1 2 ν - 1 2 μ + 1 ) ( 1 - x 2 ) - μ / 2 F ( - 1 2 ν - 1 2 μ , 1 2 ν - 1 2 μ + 1 2 ; 1 2 ; x 2 ) ,
14.3.14 w 2 ( ν , μ , x ) = 2 μ Γ ( 1 2 ν + 1 2 μ + 1 ) Γ ( 1 2 ν - 1 2 μ + 1 2 ) x ( 1 - x 2 ) - μ / 2 F ( 1 2 - 1 2 ν - 1 2 μ , 1 2 ν - 1 2 μ + 1 ; 3 2 ; x 2 ) .
14.3.19 Q ν μ ( x ) = 2 ν Γ ( ν + 1 ) ( x + 1 ) μ / 2 ( x - 1 ) ( μ / 2 ) + ν + 1 F ( ν + 1 , ν + μ + 1 ; 2 ν + 2 ; 2 1 - x ) ,
3: 5.23 Approximations
See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of Γ ( z ) . …
4: 15.14 Integrals
15.14.1 0 x s - 1 F ( a , b c ; - x ) d x = Γ ( s ) Γ ( a - s ) Γ ( b - s ) Γ ( a ) Γ ( b ) Γ ( c - s ) , min ( a , b ) > s > 0 .
5: 16.2 Definition and Analytic Properties
16.2.5 F q p ( a ; b ; z ) = F q p ( a 1 , , a p b 1 , , b q ; z ) / ( Γ ( b 1 ) Γ ( b q ) ) = k = 0 ( a 1 ) k ( a p ) k Γ ( b 1 + k ) Γ ( b q + k ) z k k ! ;
6: 14.23 Values on the Cut
14.23.3 Q ν μ ( x ± i 0 ) = e ν π i / 2 π 3 / 2 ( 1 - x 2 ) μ / 2 2 ν + 1 ( x F ( 1 2 μ - 1 2 ν + 1 2 , 1 2 ν + 1 2 μ + 1 ; 3 2 ; x 2 ) Γ ( 1 2 ν - 1 2 μ + 1 2 ) Γ ( 1 2 ν + 1 2 μ + 1 2 ) i F ( 1 2 μ - 1 2 ν , 1 2 ν + 1 2 μ + 1 2 ; 1 2 ; x 2 ) Γ ( 1 2 ν - 1 2 μ + 1 ) Γ ( 1 2 ν + 1 2 μ + 1 ) ) .
7: 7.18 Repeated Integrals of the Complementary Error Function
See accompanying text
Figure 7.18.1: Repeated integrals of the scaled complementary error function 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 0 , 1 , 2 , 4 , 8 , 16 . Magnify
8: 14.1 Special Notation
x , y , τ

real variables.

F ( a , b ; c ; z )

Olver’s scaled hypergeometric function: F ( a , b ; c ; z ) / Γ ( c ) .

9: 10.22 Integrals
10.22.49 0 t μ - 1 e - a t J ν ( b t ) d t = ( 1 2 b ) ν a μ + ν Γ ( μ + ν ) F ( μ + ν 2 , μ + ν + 1 2 ; ν + 1 ; - b 2 a 2 ) , ( μ + ν ) > 0 , ( a ± i b ) > 0 ,
10.22.50 0 t μ - 1 e - a t Y ν ( b t ) d t = cot ( ν π ) ( 1 2 b ) ν Γ ( μ + ν ) ( a 2 + b 2 ) 1 2 ( μ + ν ) F ( μ + ν 2 , 1 - μ + ν 2 ; ν + 1 ; b 2 a 2 + b 2 ) - csc ( ν π ) ( 1 2 b ) - ν Γ ( μ - ν ) ( a 2 + b 2 ) 1 2 ( μ - ν ) F ( μ - ν 2 , 1 - μ - ν 2 ; 1 - ν ; b 2 a 2 + b 2 ) , μ > | ν | , ( a ± i b ) > 0 .
10.22.56 0 J μ ( a t ) J ν ( b t ) t λ d t = a μ Γ ( 1 2 ν + 1 2 μ - 1 2 λ + 1 2 ) 2 λ b μ - λ + 1 Γ ( 1 2 ν - 1 2 μ + 1 2 λ + 1 2 ) F ( 1 2 ( μ + ν - λ + 1 ) , 1 2 ( μ - ν - λ + 1 ) ; μ + 1 ; a 2 b 2 ) , 0 < a < b , ( μ + ν + 1 ) > λ > - 1 .
10.22.58 0 J ν ( a t ) J ν ( b t ) t λ d t = ( a b ) ν Γ ( ν - 1 2 λ + 1 2 ) 2 λ ( a 2 + b 2 ) ν - 1 2 λ + 1 2 Γ ( 1 2 λ + 1 2 ) F ( 2 ν + 1 - λ 4 , 2 ν + 3 - λ 4 ; ν + 1 ; 4 a 2 b 2 ( a 2 + b 2 ) 2 ) , a b , ( 2 ν + 1 ) > λ > - 1 .
10.22.64 0 J μ + 2 n + 1 ( a t ) J μ ( b t ) d t = { b μ Γ ( μ + n + 1 ) a μ + 1 n ! F ( - n , μ + n + 1 ; μ + 1 ; b 2 a 2 ) , 0 < b < a , ( - 1 ) n / ( 2 a ) , b = a ( > 0 ) , 0 , 0 < a < b .
10: 30.15 Signal Analysis
§30.15(i) Scaled Spheroidal Wave Functions
§30.15(iv) Orthogonality
§30.15(v) Extremal Properties