In “Hydrodynamic and Hydromagnetic stability” by S. Chandrasekhar we find the analytical solutions for the Kelvin-Helmholtz instability problem for a semi-infinite heterogeneous inviscid fluid in wich density is exponentially decreasing with the height and the streaming velocity is continuously variable as a function of height . In this question, the Whittaker function and also the solutions of the Whittaker function operator equalized with the Dirac distribution or the derivative of the Dirac distribution needed to be investigated. How is mentioned by Chandrasekhar, this was been done by Dyson and Case in F.D. Dyson “Stability of an idealized atmosphere II. Zeros of the confluent hypergeometric function”- The Physics of fluids ,3, 155-7 (1960) and K.M. Case “Stability of an idealized atmosphere I. Discussion of results”, ibid. 149-54 (1960).
However, I had no access to Dyson’s and Case’s papers so here are the results of my own investigation.