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Olver hypergeometric function

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1: 15.1 Special Notation
2: 15.6 Integral Representations
15.6.1 F ( a , b ; c ; z ) = 1 Γ ( b ) Γ ( c - b ) 0 1 t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c > b > 0 .
15.6.2 F ( a , b ; c ; z ) = Γ ( 1 + b - c ) 2 π i Γ ( b ) 0 ( 1 + ) t b - 1 ( t - 1 ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c - b 1 , 2 , 3 , , b > 0 .
15.6.6 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( - t ) Γ ( c + t ) ( - z ) t d t , | ph ( - z ) | < π ; a , b 0 , - 1 , - 2 , .
15.6.7 F ( a , b ; c ; z ) = 1 2 π i Γ ( a ) Γ ( b ) Γ ( c - a ) Γ ( c - b ) - i i Γ ( a + t ) Γ ( b + t ) Γ ( c - a - b - t ) Γ ( - t ) ( 1 - z ) t d t , | ph ( 1 - z ) | < π ; a , b , c - a , c - b 0 , - 1 , - 2 , .
15.6.8 F ( a , b ; c ; z ) = 1 Γ ( c - d ) 0 1 F ( a , b ; d ; z t ) t d - 1 ( 1 - t ) c - d - 1 d t , | ph ( 1 - z ) | < π ; c > d > 0 .
3: 13.1 Special Notation
The main functions treated in this chapter are the Kummer functions M ( a , b , z ) and U ( a , b , z ) , Olver’s function M ( a , b , z ) , and the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) . …
4: 14.3 Definitions and Hypergeometric Representations
is Olver’s hypergeometric function15.1). …
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 ) ,
5: 15.7 Continued Fractions
15.7.1 F ( a , b ; c ; z ) F ( a , b + 1 ; c + 1 ; z ) = t 0 - u 1 z t 1 - u 2 z t 2 - u 3 z t 3 - ,
6: 15.15 Sums
15.15.1 F ( a , b c ; 1 z ) = ( 1 - z 0 z ) - a s = 0 ( a ) s s ! F ( - s , b c ; 1 z 0 ) ( 1 - z z 0 ) - s .
7: 15.2 Definitions and Analytical Properties
15.2.2 F ( a , b ; c ; z ) = s = 0 ( a ) s ( b ) s Γ ( c + s ) s ! z s , | z | < 1 ,
15.2.3 F ( a , b c ; x + i 0 ) - F ( a , b c ; x - i 0 ) = 2 π i Γ ( a ) Γ ( b ) ( x - 1 ) c - a - b F ( c - a , c - b c - a - b + 1 ; 1 - x ) , x > 1 .
15.2.3_5 lim c - n F ( a , b ; c ; z ) Γ ( c ) = F ( a , b ; - n ; z ) = ( a ) n + 1 ( b ) n + 1 ( n + 1 ) ! z n + 1 F ( a + n + 1 , b + n + 1 ; n + 2 ; z ) , n = 0 , 1 , 2 , .
8: 13.10 Integrals
13.10.1 M ( a , b , z ) d z = 1 a - 1 M ( a - 1 , b - 1 , z ) ,
13.10.3 0 e - z t t b - 1 M ( a , c , k t ) d t = Γ ( b ) z - b F 1 2 ( a , b ; c ; k / z ) , b > 0 , z > max ( k , 0 ) ,
13.10.5 0 e - t t b - 1 M ( a , c , t ) d t = Γ ( b ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) , ( c - a ) > b > 0 ,
13.10.10 0 t λ - 1 M ( a , b , - t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( a ) Γ ( b - λ ) , 0 < λ < a ,
13.10.16 0 e - t t 1 2 ν U ( a , b , t ) J ν ( 2 x t ) d t = Γ ( ν - b + 2 ) x 1 2 ν e - x M ( a , a - b + ν + 2 , x ) , x > 0 , max ( b - 2 , - 1 ) < ν .
9: 13.4 Integral Representations
13.4.1 M ( a , b , z ) = 1 Γ ( a ) Γ ( b - a ) 0 1 e z t t a - 1 ( 1 - t ) b - a - 1 d t , b > a > 0 ,
13.4.2 M ( a , b , z ) = 1 Γ ( b - c ) 0 1 M ( a , c , z t ) t c - 1 ( 1 - t ) b - c - 1 d t , b > c > 0 ,
13.4.3 M ( a , b , - z ) = z 1 2 - 1 2 b Γ ( a ) 0 e - t t a - 1 2 b - 1 2 J b - 1 ( 2 z t ) d t , a > 0 .
13.4.9 M ( a , b , z ) = Γ ( 1 + a - b ) 2 π i Γ ( a ) 0 ( 1 + ) e z t t a - 1 ( t - 1 ) b - a - 1 d t , b - a 1 , 2 , 3 , , a > 0 .
10: 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 .