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

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

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:

  x=0; y=-3;
  {x^y, E^(y*Log[x])}

  {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.

  x=-2;  y=(2+I)*Infinity;
  {x^y, E^(y*Log[x])}

  {Indeterminate, 0}

  x=5-6*I;  y=-Infinity*I;
  {x^y, E^(y*Log[x])}

  {Indeterminate, 0}

  {x^y, E^(y*Log[x])}

  {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.

  x=(1+2*I)*Infinity;  y=(-1+2*I)*Infinity;
  {x^y, E^(y*Log[x])}

  {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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