Analytic continuation of fundamental solutions to differential equations with constant coefficients

Christer O. Kiselman

Displaying similar documents to “Analytic continuation of fundamental solutions to differential equations with constant coefficients”

Zeros of Fekete polynomials

Brian Conrey, Andrew Granville, Bjorn Poonen, K. Soundararajan (2000)

Annales de l'institut Fourier

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For $p$ an odd prime, we show that the Fekete polynomial $f_{p} (t) = \sum_{a = 0}^{p - 1} (\frac{a}{p}) t^{a}$ has $\sim κ_{0} p$ zeros on the unit circle, where $0.500813 > κ_{0} > 0.500668$ . Here $κ_{0} - 1 / 2$ is the probability that the function $1 / x + 1 / (1 - x) + \sum_{n \in ℤ : n \neq 0, 1} δ_{n} / (x - n)$ has a zero in $] 0, 1 [$ , where each $δ_{n}$ is $\pm 1$ with y $1 / 2$ . In fact $f_{p} (t)$ has absolute value $\sqrt{p}$ at each primitive $p$ th root of unity, and we show that if $| f_{p} (e (2 i π (K + τ) / p)) | < ϵ \sqrt{p}$ for some $τ \in] 0, 1 [$ then there is a zero of $f$ close to this arc.

Simple exponential estimate for the number of real zeros of complete abelian integrals

Dmitri Novikov, Sergei Yakovenko (1995)

Annales de l'institut Fourier

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We show that for a generic polynomial $H = H (x, y)$ and an arbitrary differential 1-form $ω = P (x, y) d x + Q (x, y) d y$ with polynomial coefficients of degree $\leq d$ , the number of ovals of the foliation $H = const$ , which yield the zero value of the complete Abelian integral $I (t) = \oint_{H = t} ω$ , grows at most as $exp O_{H} (d)$ as $d \to \infty$ , where $O_{H} (d)$ depends only on $H$ . The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_{1} (t), \dots, f_{n} (t)$ , $t \in K ⋐ ℝ$ , be a fundamental system of...

Diagonal Padé approximants to hyperelliptic functions

Herbert Stahl (1996)

Annales de la Faculté des sciences de Toulouse : Mathématiques

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A quadratic integral equation

Roger D. Nussbaum (1980)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Estimate for the Number of Zeros of Abelian Integrals on Elliptic Curves

Mihajlova, Ana (2004)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: Primary 34C07, secondary 34C08. We obtain an upper bound for the number of zeros of the Abelian integral. The work was partially supported by contract No 15/09.05.2002 with the Shoumen University “K. Preslavski”, Shoumen, Bulgaria.

Painlevé’s theorem extended

Claudio Meneghini (2002)

Rendiconti del Seminario Matematico della Università di Padova

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When is the last degree solution of a Bézout identity nonnegative on the interval [-1,1]?

Tim N. T. Goodman, Charles A. Micchelli (2002)

RACSAM

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We give conditions such that the least degree solution of a Bézout identity is nonnegative on the interval [-1,1].