## Appendix A: derivation of the propagatorsThe propagators are evaluated by inserting the expansions (13) into Eq. (10), inverting the Fourier transformation analytically (since we fix ) and using the method of steepest descents to find the asymptotic dependence of the at large . For the one-dimensional case, a standard integral leads to with , from which, by the method of steepest descents (e.g., Mathews & Walker 1970 ) one finds In the case of diffusion (), this result is correctly normalised, i.e., . However, in general the normalisation is lost in this approach (cf. Rax & White 1992 ). Correcting for this, we find for the constants used in Eq. (17) the expressions ## Appendix B: the synchrotron brightness distributionThe expression for the density (26) can be directly integrated numerically. Alternatively, it is easily computed by expanding the factor and integrating over , which yields where and Note that for This expression recovers, for 3-dimensional diffusion (, ), the formula derived by Wilson (1975 ), and used by Valtaoja (1984 ) where © European Southern Observatory (ESO) 1997 Online publication: April 8, 1998 |