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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory.Linear differential operators of order m with smooth bounded coefficients are pseudo-differential operators of order m. The composition PQ of two pseudo-differential operators P, Q is again a pseudo-differential operator and the symbol of PQ can be calculated by using the symbols of P and Q. The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator.