Formulas and Theorems for the Special Functions of Mathematical Physics

Geometry of 2D topological field theories

“…b i e z dp h s i = 1 2 i I a i pe z d : (6:88) Here 1 F 1 (a; b; z) is the Kummer con uent h ypergeometric function (see [100]). We leave the proof as an exercise for the reader (see [44]).…”

The Dirac-Oscillator Green's Function

“…where Γ is the gamma function, W are the Whittaker functions of the first and second kind, respectively [11]. ( ) r r < > is the smaller (larger) of r and r′ .…”

“…Using the differential properties of the Whittaker functions [11] we obtain relations (A1) and (A2) in the Appendix, which when used in Eq. (23) give ( 2 2) ( , , ) ( Finally, one can easily verify that the relativistic bound states energy spectrum of the Dirac-Oscillator [5,9] is located at the energy poles of these components of the Green's function.…”

“…The following are some useful relations which could be obtained by using the differential formulas and recurrence relations of the Whittaker functions [11]:…”

Bessel−Zernike Discrete Variable Representation Basis

“…2 A particularly useful DVR basis set on the radial half-line is based upon the Fourier-Bessel, or more generally the Dini, representation of a function. 3 Solutions to the radial Schrodinger equation and the closely related Hankel transform may be e¤ectively expressed in terms of this representation. 4 Although these series expansions are not generally …nite, it has been conjectured that a relationship to an orthogonal (…nite) polynomial set should exist for this DVR basis set.…”