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1: 19.15 Advantages of Symmetry
β–Ί(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …
2: 19.7 Connection Formulas
β–Ί
§19.7(iii) Change of Parameter of Ξ  ⁑ ( Ο• , Ξ± 2 , k )
β–Ί
19.7.8 Ξ  ⁑ ( Ο• , Ξ± 2 , k ) + Ξ  ⁑ ( Ο• , Ο‰ 2 , k ) = F ⁑ ( Ο• , k ) + c ⁒ R C ⁑ ( ( c 1 ) ⁒ ( c k 2 ) , ( c Ξ± 2 ) ⁒ ( c Ο‰ 2 ) ) , Ξ± 2 ⁒ Ο‰ 2 = k 2 .
β–Ί
19.7.9 ( k 2 Ξ± 2 ) ⁒ Ξ  ⁑ ( Ο• , Ξ± 2 , k ) + ( k 2 Ο‰ 2 ) ⁒ Ξ  ⁑ ( Ο• , Ο‰ 2 , k ) = k 2 ⁒ F ⁑ ( Ο• , k ) Ξ± 2 ⁒ Ο‰ 2 ⁒ c 1 ⁒ R C ⁑ ( c ⁒ ( c k 2 ) , ( c Ξ± 2 ) ⁒ ( c Ο‰ 2 ) ) , ( 1 Ξ± 2 ) ⁒ ( 1 Ο‰ 2 ) = 1 k 2 .
β–Ί
19.7.10 ( 1 Ξ± 2 ) ⁒ Ξ  ⁑ ( Ο• , Ξ± 2 , k ) + ( 1 Ο‰ 2 ) ⁒ Ξ  ⁑ ( Ο• , Ο‰ 2 , k ) = F ⁑ ( Ο• , k ) + ( 1 Ξ± 2 Ο‰ 2 ) ⁒ c k 2 ⁒ R C ⁑ ( c ⁒ ( c 1 ) , ( c Ξ± 2 ) ⁒ ( c Ο‰ 2 ) ) , ( k 2 Ξ± 2 ) ⁒ ( k 2 Ο‰ 2 ) = k 2 ⁒ ( k 2 1 ) .
3: 29.15 Fourier Series and Chebyshev Series
β–Ί
29.15.1 𝑒𝐸 2 ⁒ n m ⁑ ( z , k 2 ) = 1 2 ⁒ A 0 + p = 1 n A 2 ⁒ p ⁒ cos ⁑ ( 2 ⁒ p ⁒ Ο• ) .
β–Ί
29.15.8 𝑠𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 ) = p = 0 n A 2 ⁒ p + 1 ⁒ cos ⁑ ( ( 2 ⁒ p + 1 ) ⁒ Ο• ) .
β–Ί
29.15.13 𝑐𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 ) = p = 0 n B 2 ⁒ p + 1 ⁒ sin ⁑ ( ( 2 ⁒ p + 1 ) ⁒ Ο• ) .
β–Ί
29.15.18 𝑑𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 ) = dn ⁑ ( z , k ) ⁒ ( 1 2 ⁒ C 0 + p = 1 n C 2 ⁒ p ⁒ cos ⁑ ( 2 ⁒ p ⁒ Ο• ) ) .
β–Ί
29.15.23 𝑠𝑐𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) = p = 0 n B 2 ⁒ p + 2 ⁒ sin ⁑ ( ( 2 ⁒ p + 2 ) ⁒ Ο• ) .
4: 19.21 Connection Formulas
β–Ί
§19.21(iii) Change of Parameter of R J
β–ΊChange-of-parameter relations can be used to shift the parameter p of R J from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). …
5: 8.18 Asymptotic Expansions of I x ⁑ ( a , b )
β–Ί
8.18.4 a ⁒ F k + 1 = ( k + b a ⁒ ξ ) ⁒ F k + k ⁒ ξ ⁒ F k 1 ,
β–Ί
8.18.6 ( 1 e t t ) b 1 = k = 0 d k ⁒ ( t ξ ) k .
6: 29.6 Fourier Series
β–Ί
29.6.1 𝐸𝑐 Ξ½ 2 ⁒ m ⁑ ( z , k 2 ) = 1 2 ⁒ A 0 + p = 1 A 2 ⁒ p ⁒ cos ⁑ ( 2 ⁒ p ⁒ Ο• ) .
β–Ί
29.6.16 𝐸𝑐 Ξ½ 2 ⁒ m + 1 ⁑ ( z , k 2 ) = p = 0 A 2 ⁒ p + 1 ⁒ cos ⁑ ( ( 2 ⁒ p + 1 ) ⁒ Ο• ) .
β–Ί
29.6.31 𝐸𝑠 Ξ½ 2 ⁒ m + 1 ⁑ ( z , k 2 ) = p = 0 B 2 ⁒ p + 1 ⁒ sin ⁑ ( ( 2 ⁒ p + 1 ) ⁒ Ο• ) .
β–Ί
29.6.46 𝐸𝑠 Ξ½ 2 ⁒ m + 2 ⁑ ( z , k 2 ) = p = 1 B 2 ⁒ p ⁒ sin ⁑ ( 2 ⁒ p ⁒ Ο• ) .
β–Ί
29.6.53 𝐸𝑠 Ξ½ 2 ⁒ m + 2 ⁑ ( z , k 2 ) = dn ⁑ ( z , k ) ⁒ p = 1 D 2 ⁒ p ⁒ sin ⁑ ( 2 ⁒ p ⁒ Ο• ) ,
7: 15.12 Asymptotic Approximations
β–Ί
15.12.5 𝐅 ⁑ ( a + Ξ» , b Ξ» c ; 1 2 1 2 ⁒ z ) = 2 ( a + b 1 ) / 2 ⁒ ( z + 1 ) ( c a b 1 ) / 2 ( z 1 ) c / 2 ⁒ ΞΆ ⁒ sinh ⁑ ΞΆ ⁒ ( Ξ» + 1 2 ⁒ a 1 2 ⁒ b ) 1 c ⁒ ( I c 1 ⁑ ( ( Ξ» + 1 2 ⁒ a 1 2 ⁒ b ) ⁒ ΞΆ ) ⁒ ( 1 + O ⁑ ( Ξ» 2 ) ) + I c 2 ⁑ ( ( Ξ» + 1 2 ⁒ a 1 2 ⁒ b ) ⁒ ΞΆ ) 2 ⁒ Ξ» + a b ⁒ ( ( c 1 2 ) ⁒ ( c 3 2 ) ⁒ ( 1 ΞΆ coth ⁑ ΞΆ ) + 1 2 ⁒ ( 2 ⁒ c a b 1 ) ⁒ ( a + b 1 ) ⁒ tanh ⁑ ( 1 2 ⁒ ΞΆ ) + O ⁑ ( Ξ» 2 ) ) ) ,
β–Ί
15.12.9 ( z + 1 ) 3 ⁒ Ξ» / 2 ⁒ ( 2 ⁒ Ξ» ) c 1 ⁒ 𝐅 ⁑ ( a + Ξ» , b + 2 ⁒ Ξ» c ; z ) = Ξ» 1 / 3 ⁒ ( e Ο€ ⁒ i ⁒ ( a c + Ξ» + ( 1 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) + e Ο€ ⁒ i ⁒ ( c a Ξ» ( 1 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) ) ⁒ ( a 0 ⁑ ( ΞΆ ) + O ⁑ ( Ξ» 1 ) ) + Ξ» 2 / 3 ⁒ ( e Ο€ ⁒ i ⁒ ( a c + Ξ» + ( 2 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) + e Ο€ ⁒ i ⁒ ( c a Ξ» ( 2 / 3 ) ) ⁒ Ai ⁑ ( e 2 ⁒ Ο€ ⁒ i / 3 ⁒ Ξ» 2 / 3 ⁒ Ξ² 2 ) ) ⁒ ( a 1 ⁑ ( ΞΆ ) + O ⁑ ( Ξ» 1 ) ) ,
β–Ί
15.12.13 G 0 ⁑ ( ± Ξ² ) = ( 2 + e ± ΞΆ ) c b ( 1 / 2 ) ⁒ ( 1 + e ± ΞΆ ) a c + ( 1 / 2 ) ⁒ ( z 1 e ± ΞΆ ) a + ( 1 / 2 ) ⁒ Ξ² e ΞΆ e ΞΆ .
8: 30.2 Differential Equations
β–ΊWith ΞΆ = Ξ³ ⁒ z Equation (30.2.1) changes to …
9: 31.2 Differential Equations
β–Ί
31.2.8 d 2 w d ΢ 2 + ( ( 2 ⁒ γ 1 ) ⁒ cn ⁑ ΢ ⁒ dn ⁑ ΢ sn ⁑ ΢ ( 2 ⁒ δ 1 ) ⁒ sn ⁑ ΢ ⁒ dn ⁑ ΢ cn ⁑ ΢ ( 2 ⁒ ϡ 1 ) ⁒ k 2 ⁒ sn ⁑ ΢ ⁒ cn ⁑ ΢ dn ⁑ ΢ ) ⁒ d w d ΢ + 4 ⁒ k 2 ⁒ ( α ⁒ β ⁒ sn 2 ⁑ ΢ q ) ⁒ w = 0 .
10: 2.8 Differential Equations with a Parameter
β–Ί
2.8.3 d 2 W d ΞΎ 2 = ( u 2 ⁒ z Λ™ 2 ⁒ f ⁑ ( z ) + ψ ⁑ ( ΞΎ ) ) ⁒ W ,
β–Ί
2.8.8 d 2 W / d ξ 2 = ( u 2 ⁒ ξ m + ψ ⁑ ( ξ ) ) ⁒ W ,
β–Ί
2.8.9 d 2 W d ξ 2 = ( u 2 ξ + ρ ξ 2 ) ⁒ W ,