# zeros of analytic functions

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## 11—20 of 40 matching pages

##### 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

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►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)).
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##### 13: Bibliography W

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The analyticity of Jacobian functions with respect to the parameter $k$
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Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
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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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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$
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Comput. Methods Funct. Theory 9 (2), pp. 579–591.
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Analytic Theory of Continued Fractions.
D. Van Nostrand Company, Inc., New York.
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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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##### 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: 16.4 Argument Unity

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►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.
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##### 16: 2.8 Differential Equations with a Parameter

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►In Case I there are no transition points in $\mathbf{D}$ and $g(z)$ is analytic.
In Case II $f(z)$ has a simple zero at ${z}_{0}$ and $g(z)$ is analytic at ${z}_{0}$.
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►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.
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►in which $\xi $ ranges over a bounded or unbounded interval or domain $\mathbf{\Delta}$, and $\psi (\xi )$ is ${C}^{\mathrm{\infty}}$ or analytic on $\mathbf{\Delta}$.
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►For the former $f(z)$ has a zero of multiplicity $\lambda =2,3,4,\mathrm{\dots}$ and $g(z)$ is analytic.
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##### 17: 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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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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Zeros of Stieltjes and Van Vleck polynomials.
Trans. Amer. Math. Soc. 252, pp. 197–204.
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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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Note on the trivial zeros of Dirichlet $L$-functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
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##### 18: 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

…##### 19: 2.7 Differential Equations

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►is one at which the coefficients $f(z)$ and $g(z)$ are analytic.
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.
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►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda}}_{j}$ is zero) they diverge.
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►Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution ${w}_{j}(z)$ can be continued analytically into any other sector.
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►In a finite or infinite interval $({a}_{1},{a}_{2})$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous.
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##### 20: 33.2 Definitions and Basic Properties

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