Confluent hypergeometric functions

A sequence of approximations, using properties of the confluent hypergeometric function, integration by steepest descents, and judicious discard of all but the dominant terms, gives one the asymptotic form... 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

To these three a priori reasons for considering the generalization to two indices a fourth may be added a posteriori. We shall encounter, in important specific cases, one or two of the rarer special functions associated with the confluent hypergeometric function. 

As discussed in [45], the solution of problem (26) is then obtained in terms of the confluent hypergeometric function, also known as Kummer function iFi[a b z], readily available in the Mathematica system [41], as ... 

It is useful to compare this ordering with the one for the limit R 0 (see Section 4.2). This idea was used in [41] to analyze the degeneracy of the states for some R values on the basis of properties of confluent hypergeometric functions. 

This has been analytically proven [105] using the properties of the confluent hypergeometric functions i E in Equation (7.1). For an isotropic harmonic oscillator in D-dimensions, it has been shown that... 

This is Kummer s differential equation whose regular solution at the origin is the confluent hypergeometric function lFl(—a + 1, 2, p). 

To find the energy eigenvalues they imposed the boundary conditions (Equation (4)), which is equivalent to finding the zeros of the confluent hypergeometric function F at po, i.e. they must solve j F j (— a + 1, 2, po) = 0. 

One of the most utilized methods for solving the radial equation is by expanding the wave function as a power series of the radial coordinate. This approach is very appropriate because the power series [1-3,30-32,34, 35] correctly represents the confluent hypergeometric function. 

As mentioned in an earlier section, solution of the CHA problem by means of the hypergeometric functions can be very difficult [1-4]. It is particularly difficult at the region of positive energies. In 1975 Suryanrayana and Weil [38] arrived at the solution found by de Groot and ten Seldam [3] and Dingle [4], They solved the problem by finding the zeros of the confluent... 

As has been known since the early studies of a spherically confined hydrogen atom [2,3]/ the required solution of Equation (10) is a confluent hypergeometric function, or Kummer M function,... 

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