Abstract
This chapter discusses the way to extend the approximations of a linear second-order differential equation into asymptotic expansions. The method applies to a singularity of any finite rank; the chapter focuses on the commonest case in applications, that is, unit rank. A confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. There are several common standard forms of confluent hypergeometric functions: (1) Kummer's (confluent hypergeometric) function, which is a solution to Kummer's differential equation; (2) Tricomi's (confluent hypergeometric) function is another solution to Kummer's equation; (3) Whittaker functions are solutions to Whittaker's equation; and (4) Coulomb wave functions are solutions to the Coulomb wave equation. Besser s equation can be regarded as a transformation of a special case of the confluent hypergeometric equation.
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