zeros of analytic functions
11—20 of 40 matching pages
11: 22.2 Definitions
§22.2 Definitions… ►Each is meromorphic in for fixed , with simple poles and simple zeros, and each is meromorphic in for fixed . … … ►The Jacobian functions are related in the following way. … ►In terms of Neville’s theta functions (§20.1) …
… ►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)). …
13: Bibliography W
The analyticity of Jacobian functions with respect to the parameter
Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
The zeros of Euler’s psi function and its derivatives.
J. Math. Anal. Appl. 332 (1), pp. 607–616.
The distribution of the zeros of Jacobian elliptic functions with respect to the parameter
Comput. Methods Funct. Theory 9 (2), pp. 579–591.
Analytic Theory of Continued Fractions.
D. Van Nostrand Company, Inc., New York.
On the zeros of a confluent hypergeometric function.
Proc. Amer. Math. Soc. 16 (2), pp. 281–283.
§28.7 Analytic Continuation of Eigenvalues►As functions of , and can be continued analytically in the complex -plane. …In consequence, the functions can be defined uniquely by introducing suitable cuts in the -plane. … ►All the , , can be regarded as belonging to a complete analytic function (in the large). Therefore is irreducible, in the sense that it cannot be decomposed into a product of entire functions that contain its zeros; see Meixner et al. (1980, p. 88). …
… ►The function is analytic in the parameters when its series expansion converges and the bottom parameters are not negative integers or zero. …
… ►In Case I there are no transition points in and is analytic. In Case II has a simple zero at and is analytic at . … ►The transformation is now specialized in such a way that: (a) and are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting (or part of ) has solutions that are functions of a single variable. … ►in which ranges over a bounded or unbounded interval or domain , and is or analytic on . … ►For the former has a zero of multiplicity and is analytic. …
Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable.
2nd edition, McGraw-Hill Book Co., New York.
On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations.
Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
Zeros of Stieltjes and Van Vleck polynomials.
Trans. Amer. Math. Soc. 252, pp. 197–204.
Hypergeometric Functions and Elliptic Integrals.
In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.),
Note on the trivial zeros of Dirichlet -functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
§8.2 Definitions and Basic Properties… ►Normalized functions are: … ►
§8.2(ii) Analytic Continuation… ►(8.2.9) also holds when is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. … ►
§8.2(iii) Differential Equations…
… ►is one at which the coefficients and are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►Hence unless the series (2.7.8) terminate (in which case the corresponding is zero) they diverge. … ►Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution can be continued analytically into any other sector. … ►In a finite or infinite interval let be real, positive, and twice-continuously differentiable, and be continuous. …