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.