Also, ; that is, every eigenvalue corresponds to the eigenvalue , but it has, in general, different algebraic multiplicity, geometric multiplicity and index. However, the algebraic multiplicity may be computed as follows: The function of a linear transformation between vector spaces can be defined in a similar way according to the holomorphic functional calculus, where Banach space and Riemann surface theories play a fundamental role. In the case of finite-dimensional spaces, both theories perfectly match. Dynamical systems Now suppose a (complex) dynamical system is simply defined by the equation where is the (-dimensional) curve parametrization of an orbit on the Riemann surface of the dynamical system, whereas is an complex matrix whose elements are complex functions of a -dimensional parameter . Even if (that is, continuously depends on the parameter ) the Jordan normal form of the matrix is continuously deformed almost everywhere on but, in general, not everywhere: there is some critical submanifold of on which the Jordan form abruptly changes its structure whenever the parameter crosses or simply "travels" around it (monodromy). Such changes mean that several Jordan blocks (either belonging to different eigenvalues or not) join to a unique Jordan block, or vice versa (that is, one Jordan block splits into two or more different ones). Many aspects of bifurcation theory for both continuous and discrete dynamical systems can be interpreted with the analysis of functional Jordan matrices. From the tangent space dynamics, this means that the orthogonal decomposition of the dynamical system's phase space changes and, for example, different orbits gain periodicity, or lose it, or shift from a certain kind of periodicity to another (such as period-doubling, cfr. logistic map). In a sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of . Linear ordinary differential equations The simplest example of a dynamical system is a system of linear, constant-coefficient, ordinary differential equations; that is, let and : whose direct closed-form solution involves computation of the matrix exponential: Another way, provided the solution is restricted to the local Lebesgue space of -dimensional vector fields , is to use its Laplace transform . In this case The matrix function is called the resolvent matrix of the differential operator . It is meromorphic with respect to the complex parameter since its matrix elements are rational functions whose denominator is equal for all to . Its