Astron. Astrophys. 364, 1-16 (2000)

Appendix A: order of the singularity of

As noticed in Valageas (2000a) the exponent of the singularity of (as defined in (95)) translates into the exponent for the projected generating function . However, in the cases encountered in this article we have for both the quasi-linear and highly non-linear regimes. Then is an integer but the generating function is still singular at the points through logarithmic factors. To see this, it is convenient to take the third derivative of the relation (34) which is governed by the singularity at and diverges for (while the lower derivatives of remain finite at ). This yields:

For the integral is dominated by the values of around the point where the factor is maximum, since diverges as at this point. Thus, we obtain from (A.1):

After the change of variable we obtain:

which gives:

Finally, the integration of this relation leads to:

where we only wrote the most singular term.

© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
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