푸크스 미분방정식(Fuchsian differential equation)

개요

• 선형미분방정식\[\frac{d^n w}{dz^n} + A_1(z)\frac{d^{n-1}w}{dz^{n-1}} + \cdots + A_{n-1}(z)\frac{dw}{dz} + A_n(z)w=0\]
• 모든 특이점이 정칙특이점(regular singular points)인 경우, 이를 푸크스 미분방정식이라 한다
• 19세기에 많은 연구가 진행되었음
• 초기하 미분방정식(Hypergeometric differential equations) 은 대표적인 푸크스 미분방정식의 예이다

역사

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

메모

• Dwork family http://www.math.sunysb.edu/~cschnell/pdf/notes/picardfuchs.pdf
• Group Theory and Differential Equations University of Minnesota Lecture Notes 1959-1960 by Lawrence Markus (with permission of the author)
• http://www.ima.umn.edu/~miller/Group_Theory_and_Differential_Equations.html

관련된 항목들

• 르장드르의 타원곡선 모임
• 맴돌이군이 유한인 초기하 미분방정식에 대한 슈바르츠 목록

수학용어번역

• Fuchs - 발음사전 Forvo

에세이

• Gray, J. J. 1984. “Fuchs and the Theory of Differential Equations.” Bulletin of the American Mathematical Society 10 (1): 1–26. doi:10.1090/S0273-0979-1984-15186-3.

관련논문

• Liouvillian First Integrals of Differential Equations
  • Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
• Algebraic Relations Among Solutions of Linear Differential Equations: Fano's Theorem
  • Michael F. Singer, American Journal of Mathematics, Vol. 110, No. 1 (Feb., 1988), pp. 115-143
• Elementary and Liouvillian solutions of linear differential equations
  • M. F. Singer and J. H. Davenport, 1985
• Some Applications of Linear Groups to Differential Equations
  • Michael F. Singer and Marvin D. Tretkoff, American Journal of Mathematics, Vol. 107, No. 5 (Oct., 1985), pp. 1111-1121
• Logarithmic singularities of Fuchs equations, and a criterion for the monodromy group to be finite
  • N. V. Grigorenko, MATHEMATICAL NOTESVolume 33, Number 6, 453-454
• Liouvillian Solutions of n-th Order Homogeneous Linear Differential Equations
  • Michael F. SingerAmerican Journal of Mathematics, Vol. 103, No. 4 (Aug., 1981), pp. 661-682
• Algebraic Solutions of nth Order Linear Differential Equations
  • M. Singer, Proceedings of the Queen's University 1979 Conference on Number Theory, Queens Papers in Pure and Applied Mathematics, (54), pp. 379-420
• On Second Order Linear Differential Equations with Algebraic Solutions
  • F. Baldassari, B. Dwork, American Journal of Mathematics, Vol. 101, No. 1 (Feb., 1979), pp. 42-76

관련도서

• Linear Differential Equations and Group Theory from Riemann to Poincare
  • Jeremy J. Gray, 2008(2판)
• Lectures on algebraic solutions of hypergeometric differential equations
  • Matsuda, Michihiko, 1985