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21: 15.17 Mathematical Applications
By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …
22: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). … The number m can also be replaced by any real constant λ ( > 2 ) in the sense that ( z z 0 ) λ f ( z ) is analytic and nonvanishing at z 0 ; moreover, g ( z ) is permitted to have a single or double pole at z 0 . … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . … Assume that whether or not α = a , z 2 g ( z , α ) is analytic at z = 0 . …
23: 31.4 Solutions Analytic at Two Singularities: Heun Functions
§31.4 Solutions Analytic at Two Singularities: Heun Functions
For an infinite set of discrete values q m , m = 0 , 1 , 2 , , of the accessory parameter q , the function H ( a , q ; α , β , γ , δ ; z ) is analytic at z = 1 , and hence also throughout the disk | z | < a . … with ( s 1 , s 2 ) { 0 , 1 , a , } , denotes a set of solutions of (31.2.1), each of which is analytic at s 1 and s 2 . …
24: 15.2 Definitions and Analytical Properties
§15.2 Definitions and Analytical Properties
on the disk | z | < 1 , and by analytic continuation elsewhere. … again with analytic continuation for other values of z , and with the principal branch defined in a similar way. …
§15.2(ii) Analytic Properties
The right-hand side can be seen as an analytical continuation for the left-hand side when a approaches m . …
25: 16.2 Definition and Analytic Properties
§16.2 Definition and Analytic Properties
§16.2(ii) Case p q
§16.2(iii) Case p = q + 1
§16.2(iv) Case p > q + 1
§16.2(v) Behavior with Respect to Parameters
26: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …
§3.8(v) Zeros of Analytic Functions
Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions f ( z ) . …
§3.8(vi) Conditioning of Zeros
27: 35.2 Laplace Transform
where the integration variable 𝐗 ranges over the space 𝛀 . … Then (35.2.1) converges absolutely on the region ( 𝐙 ) > 𝐗 0 , and g ( 𝐙 ) is a complex analytic function of all elements z j , k of 𝐙 . …
28: 3.12 Mathematical Constants
can be defined analytically in numerous ways, for example, …
29: 10.34 Analytic Continuation
§10.34 Analytic Continuation
30: 14.24 Analytic Continuation
§14.24 Analytic Continuation