  The method of self-similar factor approximants is completed by defining the
approximants of odd orders, constructed from the power series with the largest
term of an odd power. It is shown that the method provides good approximations
for transcendental functions. In some cases, just a few terms in a power series
make it possible to reconstruct a transcendental function exactly. Numerical
convergence of the factor approximants is checked for several examples. A
special attention is paid to the possibility of extrapolating the behavior of
functions, with arguments tending to infinity, from the related asymptotic
series at small arguments. Applications of the method are thoroughly
illustrated by the examples of several functions, nonlinear differential
equations, and anharmonic models.
