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imaginary-modulus transformations

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1: 19.7 Connection Formulas
Imaginary-Modulus Transformation
2: 15.19 Methods of Computation
For z it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when z = e ± π i / 3 . This is because the linear transformations map the pair { e π i / 3 , e π i / 3 } onto itself. …
3: 3.11 Approximation Techniques
Laplace Transform Inversion
Numerical inversion of the Laplace transform1.14(iii)) …
Example. The Discrete Fourier Transform
is called a discrete Fourier transform pair.
The Fast Fourier Transform
4: 20.11 Generalizations and Analogs
20.11.1 G ( m , n ) = k = 0 n 1 e π i k 2 m / n ;
If both m , n are positive, then G ( m , n ) allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
20.11.2 1 n G ( m , n ) = 1 n k = 0 n 1 e π i k 2 m / n = e π i / 4 m j = 0 m 1 e π i j 2 n / m = e π i / 4 m G ( n , m ) .
With the substitutions a = q e 2 i z , b = q e 2 i z , with q = e i π τ , we have … As in §20.11(ii), the modulus k of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). …
5: 10.22 Integrals
§10.22(v) Hankel Transform
The Hankel transform (or Bessel transform) of a function f ( x ) is defined as … For collections of Hankel transforms see Erdélyi et al. (1954b, Chapter 8) and Oberhettinger (1972). The following two formulas are generalizations of the Hankel transform. …This is the Weber transform. …
6: 15.8 Transformations of Variable
§15.8 Transformations of Variable
§15.8(i) Linear Transformations
§15.8(iii) Quadratic Transformations
§15.8(v) Cubic Transformations
Ramanujan’s Cubic Transformation
7: 15.12 Asymptotic Approximations
15.12.8 α = ( 2 ln ( 1 ( z 1 z + 1 ) 2 ) ) 1 / 2 ,
with the branch chosen to be continuous and α > 0 when ( ( z 1 ) / ( z + 1 ) ) > 0 . …
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 ) ) ,
a 1 ( ζ ) = ( 1 2 G 0 ( β ) 1 2 G 0 ( β ) ) / β ,
By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for F ( a + e 1 λ , b + e 2 λ ; c + e 3 λ ; z ) can be obtained with e j = ± 1 or 0 , j = 1 , 2 , 3 . …