The natural logarithm of z can
be written as an infinite product involving iterated square roots of z. A Vieta product
is defined to be a more general infinite product involving z raised to arbitrary
fractional powers. Restricted to the unit circle, Vieta products generalize infinite
cosine products studied by Salem and others in connection with PV -numbers. Vieta
products are shown to have conformal mapping, monotonicity, and growth properties
very similar to those of the natural logarithm. By using certain properties of Eulerian
polynomials, the exponents of z in a Vieta product are shown to be unique in a
strong sense.
