  Exact expressions for the partition functions of the rigid string and
membrane at any temperature are obtained in terms of hypergeometric functions.
By using zeta function regularization methods, the results are analytically
continued and written as asymptotic sums of Riemann-Hurwitz zeta functions,
which provide very good numerical approximations with just a few first terms.
This allows to obtain systematic corrections to the results of Polchinski et
al., corresponding to the limits $T\rightarrow 0$ and $T\rightarrow \infty$ of
the rigid string, and to analyze the intermediate range of temperatures. In
particular, a way to obtain the Hagedorn temperature for the rigid membrane is
thus found.
