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Differential equations with irregular singularities

In: Reflections on Quanta, Symmetries, and Supersymmetries

Author

Listed:
  • V. S. Varadarajan

    (University of California, Department of Mathematics)

Abstract

The first global studies of differential equations with rational coefficients are those of Riemann on the hypergeometric equations. These are special cases of Fuchsian equations, or, equations with regular singularities. Their theory is essentially controlled by the monodromy action. The equations with irregular singularities tell a completely different story. Here the central fact is that formal solutions do not always converge. Their theory goes back to Fabry in 1885 who discovered the phenomenon of ramification, and the decisive developments came from Hukuhara, Levelt, Turrittin, and others. In more recent times, the ideas of Balser, Deligne, Malgrange, Ramis, Subiya, Babbitt and myself, and a host of others, have created a more modern view of irregular linear differential equations that relates them to themes in commutative algebra, linear algebraic groups, and algebraic geometry.

Suggested Citation

  • V. S. Varadarajan, 2011. "Differential equations with irregular singularities," Springer Books, in: Reflections on Quanta, Symmetries, and Supersymmetries, chapter 0, pages 179-205, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-0667-0_7
    DOI: 10.1007/978-1-4419-0667-0_7
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