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

```

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