Foundations of the Theory of Special Functions
Abstract
Many important problems of theoretical and mathematical physics lead to the differential equation 1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaga % Gaey4kaSYaaSaaaeaacuaHepaDgaacaiaacIcacaWG6bGaaiykaaqa % aiabeo8aZjaacIcacaWG6bGaaiykaaaaceWG1bGbauaacqGHRaWkda % Wcaaqaaiqbeo8aZzaaiaGaaiikaiaadQhacaGGPaaabaGaeq4Wdm3a % aWbaaSqabeaacaaIYaaaaOGaaiikaiaadQhacaGGPaaaaiaadwhacq % GH9aqpcaaIWaaaaa!4E1E! $$u'' + \frac{{\tilde \tau (z)}}{{\sigma (z)}}u' + \frac{{\tilde \sigma (z)}}{{{\sigma ^2}(z)}}u = 0$$ where σ(z) and % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaG % aacaGGOaGaamOEaiaacMcaaaa!3A1E! $$\tilde \sigma (z)$$ are polynomials of degree at most 2, and % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG % aacaGGOaGaamOEaiaacMcaaaa!3A20! $$\tilde \tau (z)$$ is a polynomial of degree at most 1. Equations of this form arise, for example, in solving the Laplace and Helmholtz equations in curvilinear coordinate systems by the method of separation of variables, and in the discussion of such fundamental problems of quantum mechanics as the motion of a particle in a spherically symmetric field, the harmonic oscillator, the solution of the Schrödinger, Dirac and Klein-Gordon equations for a Coulomb potential, and the motion of a particle in a homogeneous electric or magnetic field. Moreover, equation (1) also arises in typical problems of atomic, molecular and nuclear physics.