In this book, we investigate the relative index theorem in the framework of algebraic analysis. An elliptic pair on a complex analytic manifold is the data of a coherent $\mathcal {D}$-module (i.e., a system of partial differential equations) and an $R$-constructible sheaf (for example, a local system on a subanalytic subset) satisfying some transversality condition. A natural problem is to find conditions under which the complex of holomorphic solutions of such a pair has finite-dimensional cohomology and then to compute the corresponding Euler-Poincaré characteristic. Here, we solve a relative version of this problem. We give finiteness and duality theorems unifying and extending ical results for coherent $\mathcal {O}$-modules (e.g., Grauert's theorem) or coherent $\mathcal {D}$-modules, as well as for elliptic systems. Then, to an elliptic pair, we attach a characteristic cohomology on the cotangent bundle and prove it behaves well under direct and inverse images. In particular, we establish a new kind of index formula.