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where J2j is the modified Bessel's function and the conditions take place:

The latter condition may occur and be too restrictive in actual physical problems. To eliminate the restriction determined by Eq. 2.14.23 we use the analytical continuation to the complex plane by setting Z = chx. As a result we have rr i+1/2 (2kr)M^ 1+1/2 (2kr0) = -r(2l + 2)r(/ -l)exp(r(/ -l)))° X^Zexp(-k(r + roZ))-^^ J2l+{2^Z -1)], (2.14.24)

where the contour of integration goes round the point Z = +1 in the positive direction and tends to infinity along the real semi-axis Z > 0.

Using Eq. 2.14.24 we obtain the expression for the Green's function

m^j rr0

X (1+)Zexp(- k(r + ro )Z)) -1)+1/2 J2l+{2kjrr0(Z2 - 1)] (2.14.25)

of the radial part of Eq. 2.14.12.

Notice that the Green's function determined by Eq. 2.14.25 is symmetric relative to its arguments. At l = 0, Eq. 2.14.21 and Eq. 2.14.25 represent the Green's functions of the equation for the spherically symmetric isotropic diffusion including convection.

2.14.3. Green's function of the three-dimensional transfer equation including convection

In the three-dimensional case the Green's function of the transfer equation is determined by Eq. 2.14.14. The problem is to find a closed expression for G(r,r0;rn). Using the integral representation for the Green's function Gi(r,r0;m) (Eq. 2.14.25), we obtain

G(r,r0;^)= 2 JdZexp(-k(r + r0)Z)) „+1/2 P,e), (2.14.26) 4n2iy[rr~ (Z - 1)+1/2

where

^(Z,P,e)= S( + 1)(m -1)r(l - m -1)exp((M -1))(cosJ+1 (0),(2.14.27)

and the quantity

Thus the problem reduces to a summation of the series in Eq. 2.14.27. To do this we use the following method from the paper (Hostleger, 1964). Representing the series in Eq. 2.14.27 in the form of the sum over odd and even indices and summarizing the corresponding terms, we obtain