This paper is a survey devoted to the transformations $$\begin{aligned} C&\mapsto \frac{1}{(2\pi i)^2}\int _{\Gamma _1}\int _{\Gamma _2}f(\lambda ,\mu )\,R_{1,\,\lambda }\,C\, R_{2,\,\mu }\,{\mathrm{d}}\mu \,{\mathrm{d}}\lambda ,\\ C&\mapsto \frac{1}{2\pi i}\int _{\Gamma }g(\lambda )R_{1,\,\lambda }\,C\, R_{2,\,\lambda }\,{\mathrm{d}}\lambda , \end{aligned}$$where \(R_{1,\,(\cdot )}\) and \(R_{2,\,(\cdot )}\) 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 \(\Gamma _1\), \(\Gamma _2\), and \(\Gamma\) surround the singular sets (spectra) of the pseudo-resolvents \(R_{1,\,(\cdot )}\), \(R_{2,\,(\cdot )}\), 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.