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Coulomb wave functions with complex values of the variable and the parameters. (English) Zbl 0967.81023
Summary: The motivation for the present paper lies in the fact that the literature concerning the Coulomb wave functions $F_L(\eta,\rho)$ and $G_L(\eta,\rho)$ is a jungle in which it may be hard to find a safe way when one needs general formulas for the Coulomb wave functions with complex values of the variable $\rho$ and the parameters $L$ and $\eta$. For the Coulomb wave functions and certain linear combinations of these functions we discuss the connection with the Whittaker function, the Coulomb phase shift, Wronskians, reflection formulas $(L\to-L-1)$, integral representations, series expansions, circuital relations $(\rho\to \rho e^{\pm i\pi})$ and asymptotic formulas on a Riemann surface for the variable $\rho$. The parameters $L$ and $\eta$ are allowed to assume complex values.

81Q20Semi-classical techniques in quantum theory, including WKB and Maslov methods
33E15Other wave functions
81Q99General mathematical topics and methods in quantum theory
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