Re: Re-defining Log over it's branch cut

*To*: mathgroup at smc.vnet.net*Subject*: [mg77554] Re: Re-defining Log over it's branch cut*From*: dimitris <dimmechan at yahoo.com>*Date*: Wed, 13 Jun 2007 07:19:42 -0400 (EDT)*References*: <200706110821.EAA20128@smc.vnet.net><f4lbbn$g19$1@smc.vnet.net>

I guess you use Version 6. On 5.2 I got In[19]:= Limit[z^(s - 1) /. z -> r*Exp[(-I)*(Pi - eps)], eps -> 0, Assumptions - > r > 0] Out[19]= (-r)^(-1 + s) Dimitris / Carl Woll : > chuck009 wrote: > > >Hello Dimitris, > > > >Is there a way to have Mathematica do this substitution internally when evaluating powers such as z^s? The reason I ask is that I'm working on the contour integral expressions for Zeta and Polylog which use the Hankel contour. This contour requires the substitutions z=rExp[pi i] and z=r Exp[-pi i]. However, Mathematica assigns the "standard convention" of pi to the argument for both cases. For example if I specify: > > > >In[248]:= > >N[z^(s-1)/.z->r Exp[-Pi I]] > > > >Mathematica return an answer that is actually: > > > >Exp[(s-1)(Log[r]+pi i] > > > >and not: > > > >Exp[(s-1)(Log[r]-pi i] > > > > > >I realize that's the standard convention. Just would make my code a little less messy if I didn't have to do the expansion myself and "manually" insert the -pi i factor. > > > > > One possibility is to use Limit: > > In[45]:= Limit[z^(s - 1) /. z -> r Exp[-I (Pi - eps)], eps -> 0, > Assumptions -> r > 0] > > Out[45]= E^((-1 + s)*((-I)*Pi + Log[r])) > > Carl Woll > Wolfram Research

**References**:**Re: Re-defining Log over it's branch cut***From:*chuck009 <dmilioto@comcast.com>