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

• To: mathgroup at smc.vnet.net
• Subject: [mg77539] Re: [mg77507] Re: Re-defining Log over it's branch cut
• From: Carl Woll <carlw at wolfram.com>
• Date: Tue, 12 Jun 2007 01:25:30 -0400 (EDT)
• References: <200706110821.EAA20128@smc.vnet.net>

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>