Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen

Jank, Gerhard; Volkmann, Lutz

doi:10.1007/978-3-0348-6621-7

Cited by 132 publications

(107 citation statements)

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“…Since (72) is irreducible, it is clear that none of the coefficients a ± , c ± is a solution of the equation (72). Therefore, by theorem 5.8, we have m r, for all r in a set E with infinite logarithmic measure.…”

Section: The Proof Of Theorem 55mentioning

confidence: 94%

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Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Halburd

Korhonen

2007

J. Phys. A: Math. Theor.

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The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z + 1)w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

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Section: The Proof Of Theorem 55mentioning

confidence: 94%

“…A more comprehensive description of Nevanlinna theory can be found, for example, in [19,34,54,72]. For excellent historical accounts on the development of the theory we refer to [57,81].…”

Section: Nevanlinna Theorymentioning

confidence: 99%

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Halburd

Korhonen

2007

J. Phys. A: Math. Theor.

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“…Ä ÑÑ 2.5º (see [11,12]) Let g be a transcendental entire function, let 0 < δ < 1 4 and z be such that |z| = r and that |g(z)| = M (r, g)ν g (r) Ä ÑÑ 2.7º (see [22]) Let f (z) be a non-constant entire function of finite order.…”

Section: ä ññ 23 (Marty's Criterion)º ([19]) a Family F Of Meromorphmentioning

confidence: 99%

Further results about uniqueness of meromorphic functions sharing a set with their derivatives

Zhu

2014

Mathematica Slovaca

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“…The most inspiring theorem in this theory is the classical Wiman-Valiron theorem which reveals the local behavior of entire functions and their derivatives when |f (z)| is close to its maximum modulus at z (e.g. [3,5,9]). …”

Section: Introductionmentioning

confidence: 99%

Wiman-Valiron theorem for q-differences

Wen¹,

Ye²

2016

Ann. Acad. Sci. Fenn. Math.

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Abstract. Let q be any complex number other than 0 and 1. We first asymptotically express the logarithmic q-difference log f (qz) − log f (z) in terms of the logarithmic derivative f ′ /f for any meromorphic function f of order strictly less than 1/2. Then we show the assumption that the order strictly less than 1/2 is sharp. Finally, we prove a q-difference analogue of the Wiman-Valiron theorem for entire functions of order strictly less than 1/2.

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Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen

Cited by 132 publications

References 0 publications

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Further results about uniqueness of meromorphic functions sharing a set with their derivatives

Wiman-Valiron theorem for q-differences

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