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

• To: mathgroup at smc.vnet.net
• Subject: [mg77599] Re: Re-defining Log over it's branch cut
• From: dimitris <dimmechan at yahoo.com>
• Date: Wed, 13 Jun 2007 07:44:14 -0400 (EDT)
• References: <f4j0s3$kar$1@smc.vnet.net>

Hello.Carl Woll's solution is very smart.

Limit[z^(s - 1) /. z -> r Exp[-I (Pi -eps)], eps -> 0, Assumptions ->
r > 0]
E^((-1 + s)*((-I)*Pi + Log[r]))

But I guess he used Mathematica 6, since in 5.2 I get

In[17]:=
Limit[z^(s - 1) /. z -> r*Exp[(-I)*(Pi - eps)], eps -> 0, Assumptions -
> r > 0]

Out[17]=
(-r)^(-1 + s)

So let's see what can I do (until upgrading!).

In[18]:=
PowerExpand[Limit[z^(s - 1), z -> r*Exp[e*I]]]
% /. E^(b_Times) :> ComplexExpand[E^b]
TrigToExp[% /. e -> Pi]
TrigToExp[%% /. e -> -Pi]

Out[18]=
E^(I*e*(-1 + s))*r^(-1 + s)

Out[19]=
r^(-1 + s)*(Cos[e*(-1 + s)] + I*Sin[e*(-1 + s)])

Out[20]=
E^(I*Pi*(-1 + s))*r^(-1 + s)

Out[21]=
r^(-1 + s)/E^(I*Pi*(-1 + s))

Everything is how you desired!

Regards
Dimitris


 /  chuck009       :
> 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.