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1: 19.15 Advantages of Symmetry
(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …
2: 19.21 Connection Formulas
§19.21(iii) Change of Parameter of R J
Change-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)). …
3: 19.7 Connection Formulas
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )
4: 8.18 Asymptotic Expansions of I x ( a , b )
8.18.4 a F k + 1 = ( k + b - a ξ ) F k + k ξ F k - 1 ,
8.18.6 ( 1 - e - t t ) b - 1 = k = 0 d k ( t - ξ ) k .
5: 29.6 Fourier Series
29.6.1 Ec ν 2 m ( z , k 2 ) = 1 2 A 0 + p = 1 A 2 p cos ( 2 p ϕ ) .
29.6.8 Ec ν 2 m ( z , k 2 ) = dn ( z , k ) ( 1 2 C 0 + p = 1 C 2 p cos ( 2 p ϕ ) ) .
29.6.16 Ec ν 2 m + 1 ( z , k 2 ) = p = 0 A 2 p + 1 cos ( ( 2 p + 1 ) ϕ ) .
29.6.31 Es ν 2 m + 1 ( z , k 2 ) = p = 0 B 2 p + 1 sin ( ( 2 p + 1 ) ϕ ) .
29.6.46 Es ν 2 m + 2 ( z , k 2 ) = p = 1 B 2 p sin ( 2 p ϕ ) .
6: 29.15 Fourier Series and Chebyshev Series
29.15.8 sE 2 n + 1 m ( z , k 2 ) = p = 0 n A 2 p + 1 cos ( ( 2 p + 1 ) ϕ ) .
29.15.13 cE 2 n + 1 m ( z , k 2 ) = p = 0 n B 2 p + 1 sin ( ( 2 p + 1 ) ϕ ) .
29.15.23 scE 2 n + 2 m ( z , k 2 ) = p = 0 n B 2 p + 2 sin ( ( 2 p + 2 ) ϕ ) .
7: 15.12 Asymptotic Approximations
15.12.5 F ( a + λ , b - λ c ; 1 2 - 1 2 z ) = 2 ( a + b - 1 ) / 2 ( z + 1 ) ( c - a - b - 1 ) / 2 ( z - 1 ) c / 2 ζ sinh ζ ( λ + 1 2 a - 1 2 b ) 1 - c ( I c - 1 ( ( λ + 1 2 a - 1 2 b ) ζ ) ( 1 + O ( λ - 2 ) ) + I c - 2 ( ( λ + 1 2 a - 1 2 b ) ζ ) 2 λ + a - b ( ( c - 1 2 ) ( c - 3 2 ) ( 1 ζ - coth ζ ) + 1 2 ( 2 c - a - b - 1 ) ( a + b - 1 ) tanh ( 1 2 ζ ) + O ( λ - 2 ) ) ) ,
15.12.9 ( z + 1 ) 3 λ / 2 ( 2 λ ) c - 1 F ( a + λ , b + 2 λ c ; - z ) = λ - 1 / 3 ( e π i ( a - c + λ + ( 1 / 3 ) ) Ai ( e - 2 π i / 3 λ 2 / 3 β 2 ) + e π i ( c - a - λ - ( 1 / 3 ) ) Ai ( e 2 π i / 3 λ 2 / 3 β 2 ) ) ( a 0 ( ζ ) + O ( λ - 1 ) ) + λ - 2 / 3 ( e π i ( a - c + λ + ( 2 / 3 ) ) Ai ( e - 2 π i / 3 λ 2 / 3 β 2 ) + e π i ( c - a - λ - ( 2 / 3 ) ) Ai ( e 2 π i / 3 λ 2 / 3 β 2 ) ) ( a 1 ( ζ ) + O ( λ - 1 ) ) ,
15.12.13 G 0 ( ± β ) = ( 2 + e ± ζ ) c - b - ( 1 / 2 ) ( 1 + e ± ζ ) a - c + ( 1 / 2 ) ( z - 1 - e ± ζ ) - a + ( 1 / 2 ) β e ζ - e - ζ .
8: 30.2 Differential Equations
With ζ = γ z Equation (30.2.1) changes to …
9: 31.2 Differential Equations
31.2.8 d 2 w d ζ 2 + ( ( 2 γ - 1 ) cn ζ dn ζ sn ζ - ( 2 δ - 1 ) sn ζ dn ζ cn ζ - ( 2 ϵ - 1 ) k 2 sn ζ cn ζ dn ζ ) d w d ζ + 4 k 2 ( α β sn 2 ζ - q ) w = 0 .
10: 2.8 Differential Equations with a Parameter
2.8.3 d 2 W d ξ 2 = ( u 2 z ˙ 2 f ( z ) + ψ ( ξ ) ) W ,
2.8.8 d 2 W / d ξ 2 = ( u 2 ξ m + ψ ( ξ ) ) W ,
2.8.9 d 2 W d ξ 2 = ( u 2 ξ + ρ ξ 2 ) W ,