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MathGroup Archive 2006

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66262] Re: When is x^y = != E^(y*Log[x])
  • From: "Nagu" <thogiti at gmail.com>
  • Date: Sat, 6 May 2006 01:54:35 -0400 (EDT)
  • References: <e3f555$s5h$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

x^y = exp(y*log(x)) is the y-th power function of the complex number x
and log(x) denotes a branch of logarithm.

x^y is a multifunction and have many definitions to denote y-th power
of x.

A general form would look like,

x^y = ( exp(y*ln|x| + i y*arg(x)) )*(exp(i 2*pi*y))^k, where k is a
complex number.

If a y-th power function, x^y defined on an open set S, and is
continuous on S, then it is a branch of the y-th power function.

Note that S does not contain the point zero as its epsilon-neighborhood
has many preimages.

The principal branch can be defined as follows:
x^y = ( exp(y*ln|x| + i y*arg(x)) ), for -pi < arg(x) < pi

This definition is consistent with the definition if y is integer, it
is same as a polynomial.
If y = 1/n, then our principal branch is same as the inverse of x^n.


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