About the Project

Olver’s

AdvancedHelp

(0.004 seconds)

11—20 of 146 matching pages

11: 13.4 Integral Representations
β–Ί
13.4.1 𝐌 ⁑ ( a , b , z ) = 1 Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b a ) ⁒ 0 1 e z ⁒ t ⁒ t a 1 ⁒ ( 1 t ) b a 1 ⁒ d t , ⁑ b > ⁑ a > 0 ,
β–Ί
13.4.2 𝐌 ⁑ ( a , b , z ) = 1 Ξ“ ⁑ ( b c ) ⁒ 0 1 𝐌 ⁑ ( a , c , z ⁒ t ) ⁒ t c 1 ⁒ ( 1 t ) b c 1 ⁒ d t , ⁑ b > ⁑ c > 0 ,
β–Ί
13.4.3 𝐌 ⁑ ( a , b , z ) = z 1 2 1 2 ⁒ b Ξ“ ⁑ ( a ) ⁒ 0 e t ⁒ t a 1 2 ⁒ b 1 2 ⁒ J b 1 ⁑ ( 2 ⁒ z ⁒ t ) ⁒ d t , ⁑ a > 0 .
β–Ί
13.4.9 𝐌 ⁑ ( a , b , z ) = Ξ“ ⁑ ( 1 + a b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ 0 ( 1 + ) e z ⁒ t ⁒ t a 1 ⁒ ( t 1 ) b a 1 ⁒ d t , b a 1 , 2 , 3 , , ⁑ a > 0 .
β–Ί
13.4.12 𝐌 ⁑ ( a , c , z ) = Ξ“ ⁑ ( b ) 2 ⁒ Ο€ ⁒ i ⁒ z 1 b ⁒ ( 0 + , 1 + ) e z ⁒ t ⁒ t b ⁒ 𝐅 1 2 ⁑ ( a , b ; c ; 1 / t ) ⁒ d t , b 0 , 1 , 2 , , | ph ⁑ z | < 1 2 ⁒ Ο€ .
12: 14.19 Toroidal (or Ring) Functions
β–Ί
14.19.3 𝑸 Ξ½ 1 2 ΞΌ ⁑ ( cosh ⁑ ΞΎ ) = Ο€ 1 / 2 ⁒ ( 1 e 2 ⁒ ΞΎ ) ΞΌ e ( Ξ½ + ( 1 / 2 ) ) ⁒ ΞΎ ⁒ 𝐅 ⁑ ( ΞΌ + 1 2 , Ξ½ + ΞΌ + 1 2 ; Ξ½ + 1 ; e 2 ⁒ ΞΎ ) .
β–Ί
14.19.5 𝑸 n 1 2 m ⁑ ( cosh ⁑ ΞΎ ) = Ξ“ ⁑ ( n + 1 2 ) Ξ“ ⁑ ( n + m + 1 2 ) ⁒ Ξ“ ⁑ ( n m + 1 2 ) ⁒ 0 cosh ⁑ ( m ⁒ t ) ( cosh ⁑ ΞΎ + cosh ⁑ t ⁒ sinh ⁑ ΞΎ ) n + ( 1 / 2 ) ⁒ d t , m < n + 1 2 .
β–Ί
14.19.6 𝑸 1 2 ΞΌ ⁑ ( cosh ⁑ ΞΎ ) + 2 ⁒ n = 1 Ξ“ ⁑ ( ΞΌ + n + 1 2 ) Ξ“ ⁑ ( ΞΌ + 1 2 ) ⁒ 𝑸 n 1 2 ΞΌ ⁑ ( cosh ⁑ ΞΎ ) ⁒ cos ⁑ ( n ⁒ Ο• ) = ( 1 2 ⁒ Ο€ ) 1 / 2 ⁒ ( sinh ⁑ ΞΎ ) ΞΌ ( cosh ⁑ ΞΎ cos ⁑ Ο• ) ΞΌ + ( 1 / 2 ) , ⁑ ΞΌ > 1 2 .
β–Ί β–Ί
13: 14.24 Analytic Continuation
β–Ί
14.24.1 P Ξ½ ΞΌ ⁑ ( z ⁒ e s ⁒ Ο€ ⁒ i ) = e s ⁒ Ξ½ ⁒ Ο€ ⁒ i ⁒ P Ξ½ ΞΌ ⁑ ( z ) + 2 ⁒ i ⁒ sin ⁑ ( ( Ξ½ + 1 2 ) ⁒ s ⁒ Ο€ ) ⁒ e s ⁒ Ο€ ⁒ i / 2 cos ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ Ξ“ ⁑ ( ΞΌ Ξ½ ) ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( z ) ,
β–Ί
14.24.2 𝑸 Ξ½ ΞΌ ⁑ ( z ⁒ e s ⁒ Ο€ ⁒ i ) = ( 1 ) s ⁒ e s ⁒ Ξ½ ⁒ Ο€ ⁒ i ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( z ) ,
β–Ί
14.24.4 𝑸 Ξ½ , s ΞΌ ⁑ ( z ) = e s ⁒ ΞΌ ⁒ Ο€ ⁒ i ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( z ) Ο€ ⁒ i ⁒ sin ⁑ ( s ⁒ ΞΌ ⁒ Ο€ ) sin ⁑ ( ΞΌ ⁒ Ο€ ) ⁒ Ξ“ ⁑ ( Ξ½ ΞΌ + 1 ) ⁒ P Ξ½ ΞΌ ⁑ ( z ) ,
14: 15.14 Integrals
β–Ί
15.14.1 0 x s 1 ⁒ 𝐅 ⁑ ( a , b c ; x ) ⁒ d x = Ξ“ ⁑ ( s ) ⁒ Ξ“ ⁑ ( a s ) ⁒ Ξ“ ⁑ ( b s ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c s ) , min ⁑ ( ⁑ a , ⁑ b ) > ⁑ s > 0 .
15: 14.4 Graphics
16: 15.2 Definitions and Analytical Properties
β–Ί
15.2.2 𝐅 ⁑ ( a , b ; c ; z ) = s = 0 ( a ) s ⁒ ( b ) s Ξ“ ⁑ ( c + s ) ⁒ s ! ⁒ z s , | z | < 1 ,
β–Ί
15.2.3 𝐅 ⁑ ( a , b c ; x + i ⁒ 0 ) 𝐅 ⁑ ( a , b c ; x i ⁒ 0 ) = 2 ⁒ Ο€ ⁒ i Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ ( x 1 ) c a b ⁒ 𝐅 ⁑ ( c a , c b c a b + 1 ; 1 x ) , x > 1 .
β–Ί
15.2.3_5 lim c n F ⁑ ( a , b ; c ; z ) Ξ“ ⁑ ( 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 , .
17: 14.12 Integral Representations
β–Ί
14.12.9 𝑸 n m ⁑ ( x ) = 1 n ! ⁒ 0 u ( x ( x 2 1 ) 1 / 2 ⁒ cosh ⁑ t ) n ⁒ cosh ⁑ ( m ⁒ t ) ⁒ d t ,
β–Ί
14.12.11 𝑸 n m ⁑ ( x ) = ( x 2 1 ) m / 2 2 n + 1 ⁒ n ! ⁒ 1 1 ( 1 t 2 ) n ( x t ) n + m + 1 ⁒ d t ,
β–Ί
14.12.12 𝑸 n m ⁑ ( x ) = 1 ( n m ) ! ⁒ P n m ⁑ ( x ) ⁒ x d t ( t 2 1 ) ⁒ ( P n m ⁑ ( t ) ) 2 , n m .
β–Ί
14.12.13 𝑸 n ⁑ ( x ) = 1 2 ⁒ ( n ! ) ⁒ 1 1 P n ⁑ ( t ) x t ⁒ d t .
β–Ί
18: 15.6 Integral Representations
β–Ί
15.6.1 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c b ) ⁒ 0 1 t b 1 ⁒ ( 1 t ) c b 1 ( 1 z ⁒ t ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; ⁑ c > ⁑ b > 0 .
β–Ί
15.6.2 𝐅 ⁑ ( a , b ; c ; z ) = Ξ“ ⁑ ( 1 + b c ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( b ) ⁒ 0 ( 1 + ) t b 1 ⁒ ( t 1 ) c b 1 ( 1 z ⁒ t ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; c b 1 , 2 , 3 , , ⁑ b > 0 .
β–Ί
15.6.2_5 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c b ) ⁒ 0 t b 1 ⁒ ( t + 1 ) a c ( t z ⁒ t + 1 ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; ⁑ c > ⁑ b > 0 .
β–Ί
15.6.3 𝐅 ⁑ ( a , b ; c ; z ) = e b ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( c b ) ⁒ ( 0 + ) t b 1 ⁒ ( t + 1 ) a c ( t z ⁒ t + 1 ) a ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; b 1 , 2 , 3 , , ⁑ ( c b ) > 0 .
β–Ί
15.6.8 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( c d ) ⁒ 0 1 𝐅 ⁑ ( a , b ; d ; z ⁒ t ) ⁒ t d 1 ⁒ ( 1 t ) c d 1 ⁒ d t , | ph ⁑ ( 1 z ) | < Ο€ ; ⁑ c > ⁑ d > 0 .
19: 14.23 Values on the Cut
β–Ί
14.23.2 𝑸 Ξ½ ΞΌ ⁑ ( x ± i ⁒ 0 ) = e ± ΞΌ ⁒ Ο€ ⁒ i / 2 Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( 𝖰 Ξ½ ΞΌ ⁑ ( x ) βˆ“ 1 2 ⁒ Ο€ ⁒ i ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) ) .
β–Ί
14.23.3 𝑸 Ξ½ ΞΌ ⁑ ( x ± i ⁒ 0 ) = e βˆ“ Ξ½ ⁒ Ο€ ⁒ i / 2 ⁒ Ο€ 3 / 2 ⁒ ( 1 x 2 ) ΞΌ / 2 2 Ξ½ + 1 ⁒ ( x ⁒ 𝐅 ⁑ ( 1 2 ⁒ ΞΌ 1 2 ⁒ Ξ½ + 1 2 , 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ; 3 2 ; x 2 ) Ξ“ ⁑ ( 1 2 ⁒ Ξ½ 1 2 ⁒ ΞΌ + 1 2 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) βˆ“ i ⁒ 𝐅 ⁑ ( 1 2 ⁒ ΞΌ 1 2 ⁒ Ξ½ , 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ; 1 2 ; x 2 ) Ξ“ ⁑ ( 1 2 ⁒ Ξ½ 1 2 ⁒ ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ) ) .
β–Ί
14.23.5 𝖰 Ξ½ ΞΌ ⁑ ( x ) = 1 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( e ΞΌ ⁒ Ο€ ⁒ i / 2 ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( x + i ⁒ 0 ) + e ΞΌ ⁒ Ο€ ⁒ i / 2 ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( x i ⁒ 0 ) ) ,
β–Ί
14.23.6 𝖰 Ξ½ ΞΌ ⁑ ( x ) = e βˆ“ ΞΌ ⁒ Ο€ ⁒ i / 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( x ± i ⁒ 0 ) ± 1 2 ⁒ Ο€ ⁒ i ⁒ e ± ΞΌ ⁒ Ο€ ⁒ i / 2 ⁒ P Ξ½ ΞΌ ⁑ ( x ± i ⁒ 0 ) .
20: 13.2 Definitions and Basic Properties
β–Ί
13.2.3 𝐌 ⁑ ( a , b , z ) = s = 0 ( a ) s Ξ“ ⁑ ( b + s ) ⁒ s ! ⁒ z s ,
β–Ί β–Ί
13.2.5 lim b n M ⁑ ( a , b , z ) Ξ“ ⁑ ( b ) = 𝐌 ⁑ ( a , n , z ) = ( a ) n + 1 ( n + 1 ) ! ⁒ z n + 1 ⁒ M ⁑ ( a + n + 1 , n + 2 , z ) .
β–Ί β–Ί
13.2.33 𝒲 ⁑ { 𝐌 ⁑ ( a , b , z ) , z 1 b ⁒ 𝐌 ⁑ ( a b + 1 , 2 b , z ) } = sin ⁑ ( Ο€ ⁒ b ) ⁒ z b ⁒ e z / Ο€ ,