r is Wronskian of the Eq. 2.14.16. Now, we start to solve the Eq. 2.14.16, i.e. to determine directly the functions ft(r) and (pi(r). The substitution of the independent variable r _ 2k and the unknown function
(r)) reduces the Eq. 2.14.16 to the canonic form of equation for Whittaker's function:
where u = u/k, ft = l(l +1). The condition of regularity at a zero point is satisfied by the solution of the Eq. 2.14.20 in the form of the Whittaker's function Mmj+1/2 (g) and the regularity at infinity is satisfied by the Whittaker's function Wul+y2(g).
Using the corresponding asymptotic expression for these functions, we calculate the Wronskian of Eq. 2.14.19 and obtain the Green's function Gl(r,ro;m) by means of Eq. 2.14.17:
where r(x) is Euler's r - function. The Eq. 2.14.21 completely solves the formulated problem of determining the radial Green's function of the Eq. 2.14.12. It is convenient for further considerations to represent the Green's function (Eq. 2.14.21) in the form of contour integral (Hostleger, 1964). For this purpose we shall use the integral representation (Ryzhik and Gradstein, M1971) for the product of Whittaker's functions included in Eq. 2.14.21:
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