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dn dT

( d2 d ^ -2 + 2q(r )T + Aecp dr dr where q(r ) =1 -¡

and making a substitution the unknown function according the relation1

we obtain the equation for Y(r,r) function dY A1T, (2u 2 |1T,

1 In the integral in the exponent, the lower limit of integration is not shown because its concrete value is inessential; in the inverse transition from Y(r,r) to the function n(r,r) the lower limit of integration disappears (a similar method was used in Vasilyev and Toptygin, 1976).

where

and the expression for Q1 is given in Eq. 2.14.3. We shall start below from the equation for the Green's function of the Eq. 2.14.8:

= AG(r,T; ro T)+ ^ - u 2 )g(f,t; ro T )+S(r - ro )S(t-t ).(2.14.10)

now consider a general method of composing the Green's function which satisfies the Eq. 2.14.10. For this purpose note that the function G depends only on the difference t-to owing to the invariance of the Eq. 2.14.10 and of the initial conditions with respect to the onset of the time scale. Therefore it is possible to write a formal expansion of this function into the Fourier integral

G(r,T;ro ) = — J G(r, ro ;w)exp(- im(-To ))dm, (2.14.11)

where the spectral representation of the Green's function G(r, ro;w) satisfies the equation

AG(r,ro;w) + —G(r,ro;w) + k2G(r,ro;w) = S(r - ro), (2.14.12) r where

As usual, in the integral Eq. 2.14.11, a substitution is assumed of the quantity w in the argument of the function G(r, ro ;w) by w + ie(e> 0)with conservation of integration along the real axis. This is connected with the fact that the multi-leafed function G(r, ro; w) at a certain place of a complex variable w at Imw > 0 being to have no discontinuities, according to the general theory (Morse and Feshbach, Ml953). Including this property, let us start to compose the Green's function for the Eq. 2.14.12. Note first of all that Eq. 2.14.12 coincides formally with Schrodinger's equation for a particle in a Coulomb field. Therefore the methods which were developed for solving Schrodinger's equation with a Coulomb potential (Bahrah and Vetchinkin, 1971), can be applied to the problem under consideration. Using a similarity of the Eq. 2.14.12 to Schrodinger's equation, we

Cosmic Ray Propagation in Space Plasmas i97

separate the variables in Eq. 2.14.12, presenting the Green's function G(r,r0;m) in the form of an expansion over proper functions of the angular part of the Laplacian

G(r, ro ;m) = £ ——Pi (cosO) (r, ro ;®), l=o 4n

where 6 is the angle between r and ro; l = 0, 1, 2, ...; Pi (cos 6) are Legendre's polynomials; Gt(r,ro;m) are Green's function for the radial equation corresponding to Eq. 2.14.12:

To obtain the Green's function Gi (r,ro;rn) consider the solutions fl (r) and pl(r) of the equation

2 9r 9r r2

where fl (r) is regular at r ^ 0 and the function pi (r) is regular at r ^^ . If the functions fl (r) and pi (r) are known, the Green's function will be determined, according to Titchmarsh (M1958), by the relation

G (rr . m) _ 6(r - ro )fl(r pi (ro) + 6(ro - r )fl (ro pi(r) (2i4i7)

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