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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 ${F}_{L}\left(\eta ,\rho \right)$ and ${G}_{L}\left(\eta ,\rho \right)$ 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 $\left(L\to -L-1\right)$, integral representations, series expansions, circuital relations $\left(\rho \to \rho {e}^{±i\pi }\right)$ and asymptotic formulas on a Riemann surface for the variable $\rho$. The parameters $L$ and $\eta$ are allowed to assume complex values.
##### MSC:
 81Q20 Semi-classical techniques in quantum theory, including WKB and Maslov methods 33E15 Other wave functions 81Q99 General mathematical topics and methods in quantum theory