This paper is a survey devoted to the transformations C↦1(2πi)2∫Γ1∫Γ2f(λ,μ)R1,λCR2,μdμdλ,C↦12πi∫Γg(λ)R1,λCR2,λdλ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} C&\mapsto \frac{1}{(2\pi i)^2}\int _{\Gamma _1}\int _{\Gamma _2}f(\lambda ,\mu )\,R_{1,\,\lambda }\,C\, R_{2,\,\mu }\,{\mathrm{d}}\mu \,{\mathrm{d}}\lambda ,\\ C&\mapsto \frac{1}{2\pi i}\int _{\Gamma }g(\lambda )R_{1,\,\lambda }\,C\, R_{2,\,\lambda }\,{\mathrm{d}}\lambda , \end{aligned}$$\end{document}where R1,(·)\documentclass[12pt]{minimal}
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\begin{document}$$R_{1,\,(\cdot )}$$\end{document} and R2,(·)\documentclass[12pt]{minimal}
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\begin{document}$$R_{2,\,(\cdot )}$$\end{document} are pseudo-resolvents acting in a Banach space, i. e., the resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions; here Γ1\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _1$$\end{document}, Γ2\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma _2$$\end{document}, and Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma$$\end{document} surround the singular sets (spectra) of the pseudo-resolvents R1,(·)\documentclass[12pt]{minimal}
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\begin{document}$$R_{1,\,(\cdot )}$$\end{document}, R2,(·)\documentclass[12pt]{minimal}
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\begin{document}$$R_{2,\,(\cdot )}$$\end{document}, and the both, respectively. 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 properties of the differential of the ordinary functional calculus.
