Re: Re-defining Log over it's branch cut
- To: mathgroup at smc.vnet.net
- Subject: [mg77498] Re: Re-defining Log over it's branch cut
- From: dimitris <dimmechan at yahoo.com>
- Date: Sun, 10 Jun 2007 07:25:51 -0400 (EDT)
- References: <f4dsng$hhu$1@smc.vnet.net>
Hi. In[11]:= Log[z] /. z -> r*Exp[0*I] Out[11]= Log[r] In[14]:= PowerExpand[Log[z] /. z -> r*Exp[2*Pi*i]] /. i -> I Out[14]= 2*I*Pi + Log[r] In[15]:= PowerExpand[Log[z] /. z -> r*Exp[(-Pi)*i]] /. i -> I Out[15]= (-I)*Pi + Log[r] In[16]:= PowerExpand[Log[z] /. z -> r*Exp[Pi*i]] /. i -> I Out[16]= I*Pi + Log[r] Cheers Dimitris / chuck009 : > 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)