When is x^y = != E^(y*Log[x])

*To*: mathgroup at smc.vnet.net*Subject*: [mg66236] When is x^y = != E^(y*Log[x])*From*: ted.ersek at tqci.net*Date*: Fri, 5 May 2006 05:02:22 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

The Mathematica documentation for Power says: For complex numbers (x^y) gives the principal value of ( E^(y*Log[x] ). This is consistent with reference books. I wanted to see where in the extended complex plane this identity applies. Also when it doesn't apply how do we determine (x^y). So consider the following: In[1]:= x=0; y=-3; {x^y, E^(y*Log[x])} Out[3]= {ComplexInfinity, Infinity} The documentation wasn't wrong in the above example because 0, -3 are not complex numbers. However, I have seen books that imply the identity above works for any (x,y). Well I can see the above identity doesn't apply when x=0 and y is negative. The above identity doesn't apply in the following cases either. In[4]:= x=-2; y=(2+I)*Infinity; {x^y, E^(y*Log[x])} Out[6]= {Indeterminate, 0} In[7]:= x=5-6*I; y=-Infinity*I; {x^y, E^(y*Log[x])} Out[9]= {Indeterminate, 0} In[10]:= x=y=Infinity; {x^y, E^(y*Log[x])} Out[11]= {ComplexInfinity, Infinity} Then you might say the above identity doesn't apply when Abs[y]==Infinity, but the above identity does apply in the next example. In[12]:= x=(1+2*I)*Infinity; y=(-1+2*I)*Infinity; {x^y, E^(y*Log[x])} Out[14]= {0, 0} Could somebody provide conditions on (x,y) that are necessary and sufficient for E^(y*Log[x]) to return the same thing as (x^y). ------------- Ted Ersek

**Follow-Ups**:**Re: When is x^y = != E^(y*Log[x])***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>