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Properties of the mappingsλ dλ are discussed; here R 1, (•) and R 2, (•) are pseudo-resolvents, i. e., resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.

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