Relations between closed string amplitudes and mixed string amplitudes at tree-level

This paper investigates the relationships between closed and mixed string amplitudes at the tree level in string theory. Through the analytic continuation of complex variables, we establish a factorization of closed string amplitudes into those involving (n − 2) open strings and a single closed string. Expressions for four-, five-, and six-point amplitudes are provided, together with systematic formulations for n-point string amplitudes. The paper addresses possible correction terms arising from integration along infinite tubes during the process of Wick rotations. The implications of our findings are discussed in certain limits.