Unremovable ‘removable’ singularities†

^†Dedicated to the memory of Professor Michael Lidov. This article owes a lot to (hot) discussions with Harley Flanders. The beautiful example is by courtesy of George Bergman.

Abstract

The article attempts to answer the question why the so-called ‘removable’ or ‘regular’ singularities in certain analytic functions cannot be removed. This problem may be understood in the frame of the generalized elementary functions (i.e. functions defined as solutions of explicit rational ODEs). Along with several known examples, the article produces a family of infinitely many functions having regular singularities. There are formulated also two open questions.

^†Dedicated to the memory of Professor Michael Lidov. This article owes a lot to (hot) discussions with Harley Flanders. The beautiful example is by courtesy of George Bergman.

Keywords:

• ODE
• removable singularities
• regular singularities

AMS Subject Classifications:

• 34A09
• 34A34
• 34A12

Notes

^†Dedicated to the memory of Professor Michael Lidov. This article owes a lot to (hot) discussions with Harley Flanders. The beautiful example is by courtesy of George Bergman.