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transformations of parameters

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1: 31.2 Differential Equations
w ( z ) = z 1 γ w 1 ( z ) satisfies (31.2.1) if w 1 is a solution of (31.2.1) with transformed parameters q 1 = q + ( a δ + ϵ ) ( 1 γ ) ; α 1 = α + 1 γ , β 1 = β + 1 γ , γ 1 = 2 γ . Next, w ( z ) = ( z 1 ) 1 δ w 2 ( z ) satisfies (31.2.1) if w 2 is a solution of (31.2.1) with transformed parameters q 2 = q + a γ ( 1 δ ) ; α 2 = α + 1 δ , β 2 = β + 1 δ , δ 2 = 2 δ . Lastly, w ( z ) = ( z a ) 1 ϵ w 3 ( z ) satisfies (31.2.1) if w 3 is a solution of (31.2.1) with transformed parameters q 3 = q + γ ( 1 ϵ ) ; α 3 = α + 1 ϵ , β 3 = β + 1 ϵ , ϵ 3 = 2 ϵ . … If z ~ = z ~ ( z ) is one of the 3 ! = 6 homographies that map to , then w ( z ) = w ~ ( z ~ ) satisfies (31.2.1) if w ~ ( z ~ ) is a solution of (31.2.1) with z replaced by z ~ and appropriately transformed parameters. …For example, w ( z ) = ( 1 z ) α w ~ ( z / ( z 1 ) ) , which arises from z ~ = z / ( z 1 ) , satisfies (31.2.1) if w ~ ( z ~ ) is a solution of (31.2.1) with z replaced by z ~ and transformed parameters a ~ = a / ( a 1 ) , q ~ = ( q a α γ ) / ( a 1 ) ; β ~ = α + 1 δ , δ ~ = α + 1 β . …
2: 20.7 Identities
§20.7(viii) Transformations of Lattice Parameter
20.7.33 ( i τ ) 1 / 2 θ 4 ( z | τ ) = exp ( i τ z 2 / π ) θ 2 ( z τ | τ ) .
3: 20.10 Integrals
§20.10(i) Mellin Transforms with respect to the Lattice Parameter
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
4: 30.13 Wave Equation in Prolate Spheroidal Coordinates
30.13.7 2 w + κ 2 w = 0 ,
transformed to prolate spheroidal coordinates ( ξ , η , ϕ ) , admits solutions …
30.13.9 d d ξ ( ( 1 ξ 2 ) d w 1 d ξ ) + ( λ + γ 2 ( 1 ξ 2 ) μ 2 1 ξ 2 ) w 1 = 0 ,
30.13.10 d d η ( ( 1 η 2 ) d w 2 d η ) + ( λ + γ 2 ( 1 η 2 ) μ 2 1 η 2 ) w 2 = 0 ,
Equation (30.13.7) for ξ ξ 0 , and subject to the boundary condition w = 0 on the ellipsoid given by ξ = ξ 0 , poses an eigenvalue problem with κ 2 as spectral parameter. …
5: 12.16 Mathematical Applications
PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …
6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
Transformations of Parameters
7: 29.18 Mathematical Applications
29.18.1 2 u + ω 2 u = 0 ,
when transformed to sphero-conal coordinates r , β , γ : …
29.18.6 d 2 u 2 d β 2 + ( h ν ( ν + 1 ) k 2 sn 2 ( β , k ) ) u 2 = 0 ,
29.18.7 d 2 u 3 d γ 2 + ( h ν ( ν + 1 ) k 2 sn 2 ( γ , k ) ) u 3 = 0 ,
The wave equation (29.18.1), when transformed to ellipsoidal coordinates α , β , γ : …
8: 11.7 Integrals and Sums
11.7.10 0 t ν 1 𝐇 ν ( t ) d t = π 2 ν + 1 Γ ( ν + 1 ) , ν > 3 2 ,
11.7.11 0 t μ ν 1 𝐇 ν ( t ) d t = Γ ( 1 2 μ ) 2 μ ν 1 tan ( 1 2 π μ ) Γ ( ν 1 2 μ + 1 ) , | μ | < 1 , ν > μ 3 2 ,
11.7.12 0 t μ ν 𝐇 μ ( t ) 𝐇 ν ( t ) d t = π Γ ( μ + ν ) 2 μ + ν Γ ( μ + ν + 1 2 ) Γ ( μ + 1 2 ) Γ ( ν + 1 2 ) , ( μ + ν ) > 0 .
§11.7(iii) Laplace Transforms
The following Laplace transforms of 𝐇 ν ( t ) require a > 0 for convergence, while those of 𝐋 ν ( t ) require a > 1 . …
9: 2.5 Mellin Transform Methods
§2.5(iii) Laplace Transforms with Small Parameters
2.5.41 I 1 ( x ) = h 1 ( 1 ) x 1 + 1 2 π i ρ i ρ + i x z Γ ( 1 z ) h 1 ( z ) d z ,
Similarly, since h 2 ( z ) can be continued analytically to a meromorphic function (when κ = 0 ) or to an entire function (when κ 0 ), we can choose ρ so that h 2 ( z ) has no poles in 1 < z ρ < 2 . …
2.5.42 I 2 ( x ) = β 0 z 1 res [ x z Γ ( 1 z ) h 2 ( z ) ] + 1 2 π i ρ i ρ + i x z Γ ( 1 z ) h 2 ( z ) d z .
For examples in which the integral defining the Mellin transform h ( z ) does not exist for any value of z , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
10: 2.3 Integrals of a Real Variable
Then … Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: …