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On solutions of differential equations with ``common zero'' at infinity

Árpád Elbert, Jaromír Vosmanský (1997)

Archivum Mathematicum

The zeros c k ( ν ) of the solution z ( t , ν ) of the differential equation z ' ' + q ( t , ν ) z = 0 are investigated when lim t q ( t , ν ) = 1 , | q ( t , ν ) - 1 | d t < and q ( t , ν ) has some monotonicity properties as t . The notion c κ ( ν ) is introduced also for κ real, too. We are particularly interested in solutions z ( t , ν ) which are “close" to the functions sin t , cos t when t is large. We derive a formula for d c κ ( ν ) / d ν and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair J ν ( t ) , Y ν ( t ) . We show the concavity of c κ ( ν ) for | ν | 1 2 and also...

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