The Experts below are selected from a list of 33 Experts worldwide ranked by ideXlab platform
S Yakovenko - One of the best experts on this subject based on the ideXlab platform.
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On the number of zeros of analytic functions in a neighborhood of a Fuchsian singular point with real spectrum
1996Co-Authors: M Roitman, S YakovenkoAbstract:Abstract. Consider a linear differential Equation of some order n with coefficients analytic in the unit disk. Assuming that the Equation has a unique Fuchsian singular point at z = 0, and all roots of the corresponding Indicial Equation are real, we establish an upper bound for the number of zeros of any solution of this Equation in any sector with the vertex at z = 0. This upper bound is in some sense linear in the magnitude of the coefficients of the Equation
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On The Number Of Zeros Of Analytic Functions In A Neighborhood Of A Fuchsian Singular Point With The Real Spectrum
1996Co-Authors: M Roitman, S YakovenkoAbstract:. Consider a linear differential Equation of some order n with coefficients analytic in the unit disk. Assuming that the Equation has a unique Fuchsian singular point at z = 0, and all roots of the corresponding Indicial Equation are real, we establsih an upper bound for the number of zeros of any solution of this Equation in any sector with the vertex at z = 0. This upper bound is in some sense linear in the magnitude of the coefficients of the Equation. 1. Introduction 1.1. Zeros of analytic functions determined by linear ordinary differential Equations: a brief survey. Let f(t) be a function analytic on some subset K ae C and known to satisfy a differential Equation Lf = 0 in a larger (open) domain U ' K, where L is a linear ordinary differential operator of the form L = a 0 (t) @ n + a 1 (t) @ n\Gamma1 + \Delta \Delta \Delta + an\Gamma1 (t) @ + an (t); @ = d dt ; (1) with the coefficients a j (t) analytic and bounded in U . In this paper we address a particular case of the f..
M Roitman - One of the best experts on this subject based on the ideXlab platform.
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On the number of zeros of analytic functions in a neighborhood of a Fuchsian singular point with real spectrum
1996Co-Authors: M Roitman, S YakovenkoAbstract:Abstract. Consider a linear differential Equation of some order n with coefficients analytic in the unit disk. Assuming that the Equation has a unique Fuchsian singular point at z = 0, and all roots of the corresponding Indicial Equation are real, we establish an upper bound for the number of zeros of any solution of this Equation in any sector with the vertex at z = 0. This upper bound is in some sense linear in the magnitude of the coefficients of the Equation
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On The Number Of Zeros Of Analytic Functions In A Neighborhood Of A Fuchsian Singular Point With The Real Spectrum
1996Co-Authors: M Roitman, S YakovenkoAbstract:. Consider a linear differential Equation of some order n with coefficients analytic in the unit disk. Assuming that the Equation has a unique Fuchsian singular point at z = 0, and all roots of the corresponding Indicial Equation are real, we establsih an upper bound for the number of zeros of any solution of this Equation in any sector with the vertex at z = 0. This upper bound is in some sense linear in the magnitude of the coefficients of the Equation. 1. Introduction 1.1. Zeros of analytic functions determined by linear ordinary differential Equations: a brief survey. Let f(t) be a function analytic on some subset K ae C and known to satisfy a differential Equation Lf = 0 in a larger (open) domain U ' K, where L is a linear ordinary differential operator of the form L = a 0 (t) @ n + a 1 (t) @ n\Gamma1 + \Delta \Delta \Delta + an\Gamma1 (t) @ + an (t); @ = d dt ; (1) with the coefficients a j (t) analytic and bounded in U . In this paper we address a particular case of the f..
John Michael Nahay - One of the best experts on this subject based on the ideXlab platform.
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Differential resolvents of minimal order and weight
Hindawi Limited, 2004Co-Authors: John Michael NahayAbstract:We will determine the number of powers of α that appear with nonzero coefficient in an α-power linear differential resolvent of smallest possible order of a univariate polynomial P(t) whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of an α-resolvent of smallest possible weight. We will then compute the Indicial Equation, apparent singularities, and Wronskian of the Cockle α-resolvent of a trinomial and finish with a related determinantal formula
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© Hindawi Publishing Corp. DIFFERENTIAL RESOLVENTS OF MINIMAL ORDER AND WEIGHT
2004Co-Authors: John Michael NahayAbstract:We will determine the number of powers of α that appear with nonzero coefficient in an α-power linear differential resolvent of smallest possible order of a univariate polynomial P(t) whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of an α-resolvent of smallest possible weight. We will then compute the Indicial Equation, apparent singularities, and Wronskian of the Cockle α-resolvent of a trinomial and finish with a related determinantal formula. 2000 Mathematics Subject Classification: 12H05, 13N15
Brian Haile - One of the best experts on this subject based on the ideXlab platform.
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Analytic solutions of n-th order differential Equations at a singular point
Texas State University, 2002Co-Authors: Brian HaileAbstract:Necessary and sufficient conditions are be given for the existence of analytic solutions of the nonhomogeneous n-th order differential Equation at a singular point. Let $L$ be a linear differential operator with coefficients analytic at zero. If $L^*$ denotes the operator conjugate to $L$, then we will show that the dimension of the kernel of $L$ is equal to the dimension of the kernel of $L^*$. Certain representation theorems from functional analysis will be used to describe the space of linear functionals that contain the kernel of $L^*$. These results will be used to derive a form of the Fredholm Alternative that will establish a link between the solvability of $Ly = g$ at a singular point and the kernel of $L^*$. The relationship between the roots of the Indicial Equation associated with $Ly=0$ and the kernel of $L^*$ will allow us to show that the kernel of $L^*$ is spanned by a set of polynomials
Paul Garrett - One of the best experts on this subject based on the ideXlab platform.
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[1.1] Frobenius ’ method Let
2013Co-Authors: Paul GarrettAbstract:Frobenius ’ method for solving u ′ ′ + b(x) x is slightly more complicated when the Indicial Equation u ′ + c(x) u = 0 (with b, c analytic near 0) x2 α(α − 1) + b(0)α + c(0) = 0 has repeated roots or roots differing by an integer. [1] An important example in which this occurs is the differential Equation in radial coordinates for spherical functions on hyperbolic n-space: u ′ ′ + (n − 1) coth r · u ′ − λ u = 0 The point 0 is a regular singular point for this Equation. The indical Equation is α(α − 1) + (n − 1)α = 0 When n = 2, the Indicial Equation has a double root 0. When n ≥ 3, the roots are 0 and 2 − n, which differ by an integer