Asymptotische Entwicklungen der konfluenten hypergeometrischen Funktionen U(a,b,z) und M(a,b,z) für große Werte von b und z. (Asymptotic expansions of confluent hypergeometric functions U(a,b,z) and M(a,b,z) for large values of b and z) (Q755951)

Asymptotic expansions are derived for the confluent hypergeometric functions U(a,b,z) and M(a,b,z). The parameter b and z are real and large. As is known, a transition point shows up when \(z\sim b\). Several publications are available for this asymptotic problem, in which parabolic cylinder functions are used to describe the phenomena. The present paper gives complete new expansions in terms of elementary functions for the case that \(y=z/b\) is fixed; \(y<1\) and \(y>1\) are treated separately. Later it is verified that the results include the cases \(b\to \infty\), z fixed and \(z\to \infty\), b fixed. Starting points for the investigations are the familiar integral representations of the Kummer functions.