On The Casimir Energy For A

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of 2N pieces of equal length, of alternating type I and type II material, and is taken to be relativistic in the sense that the velocity of sound always equals the velocity of light. By means of a new recursion formula we manage to calculate the Casimir energy for arbitrary integers N . Agreement with results obtained in earlier works on the string is found in all special cases. As basic regularization method we use the contour integration method. As a check, agreement is found with results obtained from the i function method (the Hurwitz function) in the case of low N (N = 1-4). The Casimir energy is generally negative, and the more so the larger is the value of N . We illustrate the results graphically in some cases. The generalization to finite temperature theory is also given. PACS numbers: 03.70.+k, 11.10.Gh, 11.10.Wx I. INTRODUCTION When dealing with zero ..