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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
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