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reciprocal-modulus transformation

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11: 22.7 Landen Transformations
§22.7 Landen Transformations
§22.7(i) Descending Landen Transformation
§22.7(ii) Ascending Landen Transformation
§22.7(iii) Generalized Landen Transformations
12: 16.20 Integrals and Series
Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer G -function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). …
13: 16.6 Transformations of Variable
§16.6 Transformations of Variable
Quadratic
Cubic
16.6.2 F 2 3 ( a , 2 b - a - 1 , 2 - 2 b + a b , a - b + 3 2 ; z 4 ) = ( 1 - z ) - a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a - b + 3 2 ; - 27 z 4 ( 1 - z ) 3 ) .
For Kummer-type transformations of F 2 2 functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
14: 17.18 Methods of Computation
§17.18 Methods of Computation
The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …
15: 27.17 Other Applications
§27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …
16: 19.13 Integrals of Elliptic Integrals
§19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals K ( k ) , E ( k ) , and D ( k ) see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
17: 3.9 Acceleration of Convergence
§3.9(i) Sequence Transformations
§3.9(ii) Euler’s Transformation of Series
Euler’s transformation is usually applied to alternating series. …
§3.9(iv) Shanks’ Transformation
§3.9(vi) Applications and Further Transformations
18: 32.7 Bäcklund Transformations
§32.7 Bäcklund Transformations
Then the transformationsThe quadratic transformation …The quartic transformation
19: 2.6 Distributional Methods
§2.6(ii) Stieltjes Transform
The Stieltjes transform of f ( t ) is defined by … f ( z ) being the Mellin transform of f ( t ) or its analytic continuation (§2.5(ii)). … Corresponding results for the generalized Stieltjes transformwhere f ( z ) is the Mellin transform of f or its analytic continuation. …
20: 7.14 Integrals
Fourier Transform
Laplace Transforms
7.14.2 0 e - a t erf ( b t ) d t = 1 a e a 2 / ( 4 b 2 ) erfc ( a 2 b ) , a > 0 , | ph b | < 1 4 π ,
7.14.3 0 e - a t erf b t d t = 1 a b a + b , a > 0 , b > 0 ,
Laplace Transforms