Polynomial Solutions of Linear Differential Equations
Language: English
Published by Walter De Gruyter Sep 2026, 2026
- Hardcover
- New
Seller: BuchWeltWeit Ludwig Meier e.K., Bergisch Gladbach, GermanyBuchWeltWeit Ludwig Meier e.K.
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Neuware -Linear ordinary differential equations with polynomial coefficients and their solutions in the complex plane have been an active area of mathematical research since the mid-19th century, with contributions from such prominent figures as G. Lamé, M. Bôcher, E. Heine, F. Klein, T. Stieltjes, G. D. Birkhoff, and many others. During the 19th century, interest focused mainly on second-order equations because of their importance in physics. The existence of a polynomial solution to a linear differential equation is a very special property, implying in particular the existence of a one-dimensional invariant subspace under the action of the monodromy group on the space of solutions. Polynomial solutions rarely occur for individual linear differential equations, but they typically arise within families of such equations. In this book, we discuss in detail two major topics: exactly solvable linear differential operators, closely related to the Bochner-Krall problem in orthogonal polynomials, and the (generalized) Heine-Stieltjes theory, both of which originated more than a century ago. Our main emphasis is on the study of the root asymptotics of various polynomial sequences appearing in both problems. We also establish connections between our asymptotic analysis and special classes of quadratic and higher-order differentials, as well as other objects of geometric origin. In addition, we present a wealth of numerical results and suggest many open problems for the interested reader.; Seit 1923 erscheinen in der Sammlung Tusculum maßgebende Editionen griechischer und lateinischer Werke mit deutscher Übersetzung. Die Originaltexte werden zudem eingeleitet und umfassend kommentiert; nach der neuen Konzeption bieten schließlich thematische Essays tiefere Einblicke in das Werk, seinen historischen Kontext und sein Nachleben. Die hohe wissenschaftliche Qualität der Ausgaben, gepaart mit dem leserfreundlichen Sprachstil der Einführungs- und Kommentarteile, macht jeden Tusculum-Band zu einer fundamentalen Lektüre nicht nur für Studierende, die sich zum ersten Mal einem antiken Autor nähern, und für Forschende, die spezifische Aspekte eines Werkes vertiefen möchten, sondern für alle, die sich durch vertrauenswürdige Übersetzungen einen Zugang zur Antiken Welt verschaffen wollen. In der Reihe wurden bisher über 340 Titel publiziert, alle erhältlich als Buch und Elektronisches Buch. Dadurch werden bislang vergriffene Titel und Raritäten wieder vollständig verfügbar gemacht. Zusätzlich zu der Buchreihe erscheint bei De Gruyter zum 90-jährigen Jubiläum das Elektronisches Buch-Paket Tusculum Online, eine digitale Sammlung aller von 1923 bis 2013 erschienenen Titel - eine gebührende Würdigung eines wichtigen Stücks deutscher Verlagsgeschichte. 361 pp. Englisch.…
Seller Inventory # 9783119147101
- Title
- Polynomial Solutions of Linear Differential Equations
- Author
- Boris Shapiro
- Publisher
- Walter De Gruyter Sep 2026
- Publication year
- 2026
- Condition
- Neu
- Binding
- Buch
- Language
- English
- ISBN 10
- 3119147109
- ISBN 13
- 9783119147101
- Item weight
- 680 grams
- Dimensions
- 241x176x22 mm
Linear ordinary differential equations with polynomial coefficients and their solutions in the complex plane have been an active area of mathematical research since the mid-19th century, with contributions from such prominent figures as G. Lamé, M. Bôcher, E. Heine, F. Klein, T. Stieltjes, G. D. Birkhoff, and many others. During the 19th century, interest focused mainly on second-order equations because of their importance in physics. The existence of a polynomial solution to a linear differential equation is a very special property, implying in particular the existence of a one-dimensional invariant subspace under the action of the monodromy group on the space of solutions. Polynomial solutions rarely occur for individual linear differential equations, but they typically arise within families of such equations.
In this book, we discuss in detail two major topics: exactly solvable linear differential operators, closely related to the Bochner–Krall problem in orthogonal polynomials, and the (generalized) Heine–Stieltjes theory, both of which originated more than a century ago. Our main emphasis is on the study of the root asymptotics of various polynomial sequences appearing in both problems. We also establish connections between our asymptotic analysis and special classes of quadratic and higher-order differentials, as well as other objects of geometric origin. In addition, we present a wealth of numerical results and suggest many open problems for the interested reader.
"Synopsis" may belong to another edition of this title.
About the Author
Prof. B.Shapiro has been working in Stockholm University since 1991. He has published around 130 articles in various areas of mathematics, mainly in differential equations, topology and commutative algebra. One of his (joint) achievements is the formulation of the so-called Shapiro-Shapiro conjecture in real Schubert calculus which has been later settled by Mukhin-Tarasov-Varchenko. Another is his finding of a very intriguing, but completely forgotten conjecture on the number of equilibrium points for systems of point charges in 3-dimensional space due to J.C.Maxwell which was published in his book “A Treatise on Electricity and Magnetism” in 1873.
Prof. B.Shapiro is a permanent recipient of research grants from the Swedish Science Foundation and served as an advisor for 12 doctoral students.
"About the title" may belong to another edition of this title.
BuchWeltWeit Ludwig Meier e.K.
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