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1984•SIAM Journal on Mathematical AnalysisRequires access

A Connection Problem for a Regular and an Irregular Singular Point of Complex Ordinary Differential Equations

Reinhard Schäfke

Open publisher page 19 citations

Abstract

In recent papers [SIAM J. Math. Anal., 11 (1980), pp. 848–862, 863–875] D. Schmidt and the author studied the connection problem for two neighboring regular singular points for quite general linear complex ordinary differential equations. In the present paper these results are generalized in the case that one singular point is irregular singular, but of rank 1 and the leading matrix has n distinct eigenvalues; one singular point is regular singular; and there may be several other singular points. The main result is a limit formula for those connection coefficients which are essential in some specified way. An application to the generalized Heun equation is given.

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What this paper is about

In recent papers [SIAM J. Math. Anal., 11 (1980), pp. 848–862, 863–875] D. Schmidt and the author studied the connection problem for two neighboring regular singular points for quite general linear complex ordinary differential equations. In the present paper these results are generalized in the case that one singular point is irregular singular, but of rank 1 and the leading matrix has n distinct eigenvalues; one singular point is regular singular; and there may be several other singular points. The main result is a limit formula for those connection coefficients which are essential in some specified way. An application to the generalized Heun equation is given.

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Available abstract

In recent papers [SIAM J. Math. Anal., 11 (1980), pp. 848–862, 863–875] D. Schmidt and the author studied the connection problem for two neighboring regular singular points for quite general linear complex ordinary differential equations. In the present paper these results are generalized in the case that one singular point is irregular singular, but of rank 1 and the leading matrix has n distinct eigenvalues; one singular point is regular singular; and there may be several other singular points. The main result is a limit formula for those connection coefficients which are essential in some specified way. An application to the generalized Heun equation is given.

Key concepts: Mathematics, Regular singular point, Singular solution, Singular point of a curve, Connection (principal bundle), Eigenvalues and eigenvectors, Ordinary differential equation, Mathematical analysis

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