  We establish the mapping relations between analytic functions and periodic
functions using the abstract operators $\cos(h\partial_x)$ and
$\sin(h\partial_x)$, including the mapping relations between power series and
trigonometric series, and by using such mapping relations we obtain a general
method to find the sum function of a trigonometric series. According to this
method, if each coefficient of a power series is respectively equal to that of
a trigonometric series, then if we know the sum function of the power series,
we can obtain that of the trigonometric series, and the non-analytical points
of which are also determined at the same time, thus we obtain a general method
to find the sum of the Dirichlet series of integer variables, and derive
several new properties of $\zeta(2n+1)$.
