  The Casimir energy for the transverse oscillations of a piecewise uniform
closed string is calculated. The string is relativistic in the sense that the
velocity of transverse waves is always equal to c. The great adaptibility of
this string model with respect to various regularization methods is pointed
out. We survey several regularization methods: the cutoff method, the complex
contour integration method, and the zeta-function method. The most powerful
method in the present case is the contour integration method. The Casimir
energy turns out to be negative, and the more so the larger is the number of
pieces in the string. The thermodynamic free energy F is calculated for a
two-piece string in the limit when the tension ratio x = T_I/T_II approaches
zero. For large values of the length ratio s = L_II/L_I, the Hagedorn
temperature becomes proportional to the square root of s.
