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0o Introduction. In this paper we shall give complex powers of a system of pseudo-differential operators which is not necessarily elliptic. Complex powers of an elliptic pseudo-differential operator were defined by Seeley . He constructed complex powers of a pseudodifferential operator p(x, D) defined on a compact C-manifold without boundary. Here we shall construct symbols for complex powers only by local calculation which works even for operators defined locally. Recently Nagase-Shinkaidoi:10.3792/pja/1195519968 fatcat:x6umtee5szg3lf6v4apvupu2ye