  We present an explicit formula for the discrete power function introduced by
Bobenko, which is expressed in terms of the hypergeometric \tau functions for
the sixth Painlev\'e equation. The original definition of the discrete power
function imposes strict conditions on the domain and the value of the exponent.
However, we show that one can extend the value of the exponent to arbitrary
complex numbers except even integers and the domain to a discrete analogue of
the Riemann surface. Moreover, we show that the discrete power function is an
immersion when the real part of the exponent is equal to one.
