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Re-defining Log over it's branch cut

• To: mathgroup at smc.vnet.net
• Subject: [mg77458] Re-defining Log over it's branch cut
• From: chuck009 <dmilioto at comcast.com>
• Date: Sat, 9 Jun 2007 05:33:56 -0400 (EDT)

Hello all,

When integrating along a logarithmic branch cut, often have to define two values of z:

z=r Exp[0 i] which is a path above the positive real axis 

z=r Exp[2 pi i] which is the path below the positive real axis

However, when I make a transformation rule:

Log[z] /.z->r Exp[0 i]  returns Log[r]

but:

Log[z] /.z->r Exp[2 pi i] should be Log[r]+2 pi i

however Mathematica returns Log[r] in both cases.


Is there a way to configure the Log function to return these two values?  I have the same question when the branch cut is along the negative real axis:

Log[z] /. z->r Exp[-pi i] (needs to be Log[r]-pi i)

Log[z] /. z->r Exp[pi i]  (needs to be Log[r]+pi i)