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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument
Convergence Properties
2: 3.3 Interpolation
§3.3(vi) Other Interpolation Methods
These references also describe convergence properties of the interpolation formulas. …
3: 3.10 Continued Fractions
can be converted into a continued fraction C of type (3.10.1), and with the property that the n th convergent C n = A n / B n to C is equal to the n th partial sum of the series in (3.10.3), that is, …
4: 2.5 Mellin Transform Methods
when this integral converges. … To ensure that the integral (2.5.3) converges we assume that … With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1. … The extended transform f ( z ) has the same properties as f 1 ( z ) in the half-plane z < b . … Next from Table 2.5.1 we observe that the integrals for the transform pair f j ( 1 - z ) and h k ( z ) are absolutely convergent in the domain D j k specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20). …
5: 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
6: 21.2 Definitions
21.2.1 θ ( z | Ω ) = n g e 2 π i ( 1 2 n Ω n + n z ) .
This g -tuple Fourier series converges absolutely and uniformly on compact sets of the z and Ω spaces; hence θ ( z | Ω ) is an analytic function of (each element of) z and (each element of) Ω . …
7: 1.3 Determinants
§1.3(i) Definitions and Elementary Properties
converges1.9(vii)). …Hill-type determinants always converge. …
8: 2.1 Definitions and Elementary Properties
§2.1 Definitions and Elementary Properties
Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … The asymptotic property may also hold uniformly with respect to parameters. … As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. … Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …
9: 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 . … Assume that 𝒮 | g ( U + i V ) | d V converges, and also that its limit as U is 0 . …
10: 15.2 Definitions and Analytical Properties
§15.2 Definitions and Analytical Properties
On the circle of convergence, | z | = 1 , the Gauss series:
  • (a)

    Converges absolutely when ( c - a - b ) > 0 .

  • §15.2(ii) Analytic Properties
    Because of the analytic properties with respect to a , b , and c , it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …