SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

Appendix A: order of the singularity of [FORMULA]

As noticed in Valageas (2000a) the exponent [FORMULA] of the singularity of [FORMULA] (as defined in (95)) translates into the exponent [FORMULA] for the projected generating function [FORMULA]. However, in the cases encountered in this article we have [FORMULA] for both the quasi-linear and highly non-linear regimes. Then [FORMULA] is an integer but the generating function [FORMULA] is still singular at the points [FORMULA] 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 [FORMULA] and diverges for [FORMULA] (while the lower derivatives of [FORMULA] remain finite at [FORMULA]). This yields:

[EQUATION]

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

[EQUATION]

After the change of variable [FORMULA] we obtain:

[EQUATION]

which gives:

[EQUATION]

Finally, the integration of this relation leads to:

[EQUATION]

where we only wrote the most singular term.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
helpdesk.link@springer.de