case the potential energy V(x) is: This leads to the following ordinary second order differential equation: This differential equation has an irregular singularity at ∞. Its solutions are Hermite polynomials. Hypergeometric equation The equation may be defined as Dividing both sides by gives: This differential equation has regular singular points at 0, 1 and ∞. A solution is the hypergeometric function. References E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (1935) A. R. Forsyth Theory of Differential Equations Vol. IV: Ordinary Linear Equations (Cambridge University Press, 1906) Édouard Goursat, A Course in Mathematical Analysis, Volume II, Part II: Differential Equations pp. 128−ff. (Ginn & co., Boston, 1917) E. L. Ince, Ordinary Differential Equations, Dover Publications (1944) T. M. MacRobert Functions of a Complex Variable p. 243 (MacMillan, London, 1917) E. T. Whittaker and G. N. Watson A Course of Modern Analysis pp. 188−ff. (Cambridge University Press, 1915) Ordinary differential equations Complex analysis