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11: 19.7 Connection Formulas
Reciprocal-Modulus Transformation
Imaginary-Modulus Transformation
Imaginary-Argument Transformation
With sinh ϕ = tan ψ , … For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …
12: 31.2 Differential Equations
By composing these three steps, there result 2 3 = 8 possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). …
13: 35.2 Laplace Transform
where the integration variable X ranges over the space Ω . …
14: 12.16 Mathematical Applications
PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).
15: 19.8 Quadratic Transformations
§19.8 Quadratic Transformations
§19.8(ii) Landen Transformations
Descending Landen Transformation
Ascending Landen Transformation
§19.8(iii) Gauss Transformation
16: 31.10 Integral Equations and Representations
Transformation of Independent Variable
17: 12.10 Uniform Asymptotic Expansions for Large Parameter
In this section we give asymptotic expansions of PCFs for large values of the parameter a that are uniform with respect to the variable z , when both a and z ( = x ) are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables. With the transformationsThe variable ζ is defined by …
12.10.40 ϕ ( ζ ) = ( ζ t 2 - 1 ) 1 4 .
18: 1.14 Integral Transforms
1.14.1 ( f ) ( x ) = f ( x ) = 1 2 π - f ( t ) e i x t d t .
The same notation is used for Fourier transforms of functions of several variables and for Fourier transforms of distributions; see §1.16(vii). …
1.14.46 1 2 π - f ( u ) e i u x d u = - i ( sign x ) f ( x ) ,
19: 1.10 Functions of a Complex Variable
Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. …
20: 22.7 Landen Transformations
§22.7 Landen Transformations
§22.7(i) Descending Landen Transformation
§22.7(ii) Ascending Landen Transformation
k 2 = 2 k 1 + k ,
§22.7(iii) Generalized Landen Transformations