Whittaker equation
(0.002 seconds)
21—30 of 60 matching pages
21: 18.34 Bessel Polynomials
22: 23.21 Physical Applications
§23.21(ii) Nonlinear Evolution Equations
βΊAirault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). …23: 28.8 Asymptotic Expansions for Large
24: 33.14 Definitions and Basic Properties
25: 33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22(i) Schrödinger Equation
… βΊ§33.22(iv) Klein–Gordon and Dirac Equations
… βΊFor bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function . … βΊ§33.22(vi) Solutions Inside the Turning Point
… βΊ26: 28.1 Special Notation
integers. | |
… | |
real or complex parameters of Mathieu’s equation with . | |
… |
, | , | , | , | , |
27: Bibliography N
28: 2.8 Differential Equations with a Parameter
§2.8 Differential Equations with a Parameter
… βΊThe transformed equation has the form …In Case III the approximating equation is … βΊThe transformed differential equation is … βΊFor examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …29: 23.22 Methods of Computation
In the general case, given by , we compute the roots , , , say, of the cubic equation ; see §1.11(iii). These roots are necessarily distinct and represent , , in some order.
If and are real, and the discriminant is positive, that is , then , , can be identified via (23.5.1), and , obtained from (23.6.16).
If , or and are not both real, then we label , , so that the triangle with vertices , , is positively oriented and is its longest side (chosen arbitrarily if there is more than one). In particular, if , , are collinear, then we label them so that is on the line segment . In consequence, , satisfy (with strict inequality unless , , are collinear); also , .
Finally, on taking the principal square roots of and we obtain values for and that lie in the 1st and 4th quadrants, respectively, and , are given by
where denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs (, ), corresponding to the 2 possible choices of the square root.