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11: 8.11 Asymptotic Approximations and Expansions
8.11.11 γ * ( 1 - a , - x ) = x a - 1 ( - cos ( π a ) + sin ( π a ) π ( 2 π F ( y ) + 2 3 2 π a ( 1 - y 2 ) ) e y 2 + O ( a - 1 ) ) ,
12: 29.2 Differential Equations
29.2.4 ( 1 - k 2 cos 2 ϕ ) d 2 w d ϕ 2 + k 2 cos ϕ sin ϕ d w d ϕ + ( h - ν ( ν + 1 ) k 2 cos 2 ϕ ) w = 0 ,
29.2.5 ϕ = 1 2 π - am ( z , k ) .
13: 19.25 Relations to Other Functions
The three changes of parameter of Π ( ϕ , α 2 , k ) in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). …
14: 19.8 Quadratic Transformations
15: 15.8 Transformations of Variable
15.8.32 ( 1 - z 3 ) a ( - z ) 3 a ( 1 Γ ( a + 2 3 ) Γ ( 2 3 ) F ( a , a + 1 3 2 3 ; z - 3 ) + e 1 3 π i z Γ ( a ) Γ ( 4 3 ) F ( a + 1 3 , a + 2 3 4 3 ; z - 3 ) ) = 3 3 2 a + 1 2 e 1 2 a π i Γ ( a + 1 3 ) ( 1 - ζ ) a 2 π Γ ( 2 a + 2 3 ) ( - ζ ) 2 a F ( a + 1 3 , 3 a 2 a + 2 3 ; ζ - 1 ) , | z | > 1 , | ph ( - z ) | < 1 3 π .
16: 28.2 Definitions and Basic Properties
28.2.2 ζ ( 1 - ζ ) w ′′ + 1 2 ( 1 - 2 ζ ) w + 1 4 ( a - 2 q ( 1 - 2 ζ ) ) w = 0 .
28.2.3 ( 1 - ζ 2 ) w ′′ - ζ w + ( a + 2 q - 4 q ζ 2 ) w = 0 .
17: 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
Then
20.10.4 0 e - s t θ 1 ( β π 2 | i π t 2 ) d t = 0 e - s t θ 2 ( ( 1 + β ) π 2 | i π t 2 ) d t = - s sinh ( β s ) sech ( s ) ,
20.10.5 0 e - s t θ 3 ( ( 1 + β ) π 2 | i π t 2 ) d t = 0 e - s t θ 4 ( β π 2 | i π t 2 ) d t = s cosh ( β s ) csch ( s ) .
18: 3.8 Nonlinear Equations
Then the sensitivity of a simple zero z to changes in α is given by …
19: 31.7 Relations to Other Functions
31.7.1 F 1 2 ( α , β ; γ ; z ) = H ( 1 , α β ; α , β , γ , δ ; z ) = H ( 0 , 0 ; α , β , γ , α + β + 1 - γ ; z ) = H ( a , a α β ; α , β , γ , α + β + 1 - γ ; z ) .
Other reductions of H to a F 1 2 , with at least one free parameter, exist iff the pair ( a , p ) takes one of a finite number of values, where q = α β p . Below are three such reductions with three and two parameters. …
31.7.2 H ( 2 , α β ; α , β , γ , α + β - 2 γ + 1 ; z ) = F 1 2 ( 1 2 α , 1 2 β ; γ ; 1 - ( 1 - z ) 2 ) ,
With z = sn 2 ( ζ , k ) and …
20: 28.14 Fourier Series
28.14.4 q c 2 m + 2 - ( a - ( ν + 2 m ) 2 ) c 2 m + q c 2 m - 2 = 0 , a = λ ν ( q ) , c 2 m = c 2 m ν ( q ) ,
28.14.5 m = - ( c 2 m ν ( q ) ) 2 = 1 ;
For changes of sign of ν , q , and m ,
28.14.7 c - 2 m - ν ( q ) = c 2 m ν ( q ) ,
28.14.8 c 2 m ν ( - q ) = ( - 1 ) m c 2 m ν ( q ) .