Applications of Coulomb wave functions to fluid mechanics.

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Coulomb wave functions belong to a family of confluent hypergeometric functions and are mainly used in scattering theories in quantum physics. The aim of the research is to apply them to fluid mechanics. First, the convergence of an orthogonal series whose bases are Stokes stream functions of axisymmetric stationary Euler flows and represented with a Coulomb wave function is studied. As a result, a convergence theorem analogous to that of the Fourier or the Fourier-Bessel series is proved. Next, the distribution of complex phase velocities for small disturbances to pipe Poiseuille flow is analytically explained by using asymptotic forms of a Coulomb wave function with a complex variable and a complex parameter.