zeros of analytic functions

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11: 22.2 Definitions

§22.2 Definitions

… ►Each is meromorphic in $z$ for fixed $k$, with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$. … … ►The Jacobian functions are related in the following way. … ►In terms of Neville’s theta functions (§20.1) …

12: 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 $\frac{1}{3}$ (§9.6(i)). …

13: Bibliography W

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The analyticity of Jacobian functions with respect to the parameter $k$ . Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§22.17(ii), §22.2.

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The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

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ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:

ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

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Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.

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Notes:

Reprinted by Chelsea Publishing, New York, 1973 and AMS Chelsea Publishing, Providence, RI, 2000.

External Links:

ISBN 0-8218-2106-7, MathReview (R. C. Buck), ZentralBlatt

Referenced by:

§3.10(iii), §4.25, §4.9(i), §4.9(ii), Ch.4, §5.10.

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On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16 (2), pp. 281–283.

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External Links:

FullText, Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§13.9(ii).

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14: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

►As functions of $q$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be continued analytically in the complex $q$-plane. …In consequence, the functions can be defined uniquely by introducing suitable cuts in the $q$-plane. … ►All the $a_{2n}\left(q\right)$, $n=0,1,2,\mathrm{\dots }$, can be regarded as belonging to a complete analytic function (in the large). Therefore $w_{\text{I}}^{\prime }(\frac{1}{2}\pi ;a,q)$ 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). …

15: 1.4 Calculus of One Variable

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§1.4(i) Monotonicity

… ► … ►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus $C^{\mathrm{\infty }}$, and well defined for all values of these variables; possible exceptions being at boundary points. ►A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for $x\in ℝ$. … ►

§1.4(viii) Convex Functions

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16: 16.4 Argument Unity

… ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. …

17: 2.8 Differential Equations with a Parameter

… ►In Case I there are no transition points in $𝐃$ and $g(z)$ is analytic. In Case II $f(z)$ has a simple zero at $z_0$ and $g(z)$ is analytic at $z_0$. … ►The transformation is now specialized in such a way that: (a) $\xi$ and $z$ are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting $\psi (\xi )$ (or part of $\psi (\xi )$) has solutions that are functions of a single variable. … ►in which $\xi$ ranges over a bounded or unbounded interval or domain $𝚫$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $𝚫$. … ►For the former $f(z)$ has a zero of multiplicity $\lambda =2,3,4,\mathrm{\dots }$ and $g(z)$ is analytic. …

18: Bibliography

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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.

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Notes:

A third edition was published in 1978 by McGraw-Hill, New York.

External Links:

MathReview (P. Lelong), ZentralBlatt

Referenced by:

§1.9(iii), §23.18.

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:

ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

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Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§31.15(ii).

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Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

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External Links:

MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.9(i).

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Note on the trivial zeros of Dirichlet $L$-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.

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External Links:

ISSN 0002-9939, FullText, MathReview (Jerzy Kaczorowski), ZentralBlatt

Referenced by:

(25.15.7), (25.15.8), §25.15(ii), §25.15.

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19: Bibliography B

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Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§9.13(ii).

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Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.

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ISSN 1064-8275, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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External Links:

Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

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Bessel Functions. Part I: Functions of Orders Zero and Unity. Mathematical Tables, Volume 6, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-04319-0, ZentralBlatt

Referenced by:

§10.1, 1st item, 1st item, 1st item.

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An analytic continuation formula for the generalized hypergeometric function. SIAM J. Math. Anal. 19 (5), pp. 1249–1251.

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ISSN 0036-1410, Document, MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§16.8(ii).

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20: 8.2 Definitions and Basic Properties

§8.2 Definitions and Basic Properties

… ►Normalized functions are: … ►

§8.2(ii) Analytic Continuation

… ►(8.2.9) also holds when $a$ is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. … ►

§8.2(iii) Differential Equations

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