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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
Thus if b 0 0 , then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in z p + q . …
Laplace Transform Inversion
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
where ψ ( x ) = Γ ( x ) / Γ ( x ) 5.2(i)). …
§10.22(v) Hankel Transform
The Hankel transform (or Bessel transform) of a function f ( x ) is defined as … For asymptotic expansions of Hankel transforms see Wong (1976, 1977), Frenzen and Wong (1985a) and Galapon and Martinez (2014). For collections of Hankel transforms see Erdélyi et al. (1954b, Chapter 8) and Oberhettinger (1972). …
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 . …