relation to Whittaker equation
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1: 13.14 Definitions and Basic Properties
2: 31.8 Solutions via Quadratures
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โบFor , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form.
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3: 13.27 Mathematical Applications
§13.27 Mathematical Applications
… โบThe other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. … โบFor applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).4: 13.18 Relations to Other Functions
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§13.18(i) Elementary Functions
… โบ§13.18(ii) Incomplete Gamma Functions
… โบ§13.18(iii) Modified Bessel Functions
… โบ§13.18(iv) Parabolic Cylinder Functions
… โบ§13.18(v) Orthogonal Polynomials
…5: 33.16 Connection Formulas
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§33.16(i) and in Terms of and
… โบ§33.16(ii) and in Terms of and when
… โบ§33.16(iii) and in Terms of when
… โบ§33.16(iv) and in Terms of and when
… โบ§33.16(v) and in Terms of when
…6: 33.22 Particle Scattering and Atomic and Molecular Spectra
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§33.22(i) Schrödinger Equation
… โบThe relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). …The solutions to this equation are closely related to the Coulomb functions; see Greiner et al. (1985). … โบFor scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, and , or and , to determine the scattering -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). โบFor bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function . …7: 10.16 Relations to Other Functions
§10.16 Relations to Other Functions
โบElementary Functions
… โบParabolic Cylinder Functions
… โบConfluent Hypergeometric Functions
… โบGeneralized Hypergeometric Functions
…8: 12.7 Relations to Other Functions
§12.7 Relations to Other Functions
โบ§12.7(i) Hermite Polynomials
… โบ§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function
… โบ§12.7(iii) Modified Bessel Functions
… โบ§12.7(iv) Confluent Hypergeometric Functions
…9: 18.34 Bessel Polynomials
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18.34.1
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