Holomorphic integral

Theorem 9.3. If all Lyapunov values are equal to zero, then the associated analytic system has an analytic invariant (center) manifold which is filled with closed trajectories around the origin, as shown in Fig. 9.3.3. On the center manifold the system has a holomorphic integral of the type... 

We will determine a coherent-state/holomorphic path integral representation for the parallel-transport operator, deriving an appropriate transition amplitude [a path integral counterpart of the l.h.s. in Eq. (14)]. 

At present, we are prepared to formulate a holomorphic path-integral version of the non-Abelian Stokes theorem... 

The problem deals with reconstruction of a holomorphic function, where the real and imaginary parts are harmonic and satisfy CR. Therefore having one function known, the second one can be easily obtained by integration of CR. Indeed, assume, for instance, that the real part of the holomorphic function is known analytically (p x, y). CR allows finding the former by integration ... 

The recovering of current density from data on electric potential, satisfying Laplace s equation was studied. In experiments, it is difficult or expensive to obtain many measurements and therefore numerical integration cannot be performed. The recovered results revealed high accuracy with synthetic ideal function, as for ideal data, so does for data subjected to high errors. The method uses complex variable theory where one can obtain holomorphic function, related to the electric potential and its derivative related with the current density. 

The specific feature of the compelx analytic case is that on complex manifolds there are few holomorphic functions. For instance, if a manifold is compact then, by the well-known Liouville theorem, there is not a single non-constant holomorphic function on this manifold. Therefore there is no point in literally transferring the standard definition of the completely integrable symplectic structure from the smooth case. We shall further analyze several distinct notions of Liouville integrability. This question has been examined by Markushevich, and the results are presented below. 

The assertion of item (3) follows from the holomorphic version of the Liouville theorem on integrable systems, see 49 of the book [3j. The proof of the holomorphic version is quite similar to the standard one. 

The Holomorphic l Form of the Integral of a Geodesic Flow Polynomial in Momenta and the Theorem on Nonintegrability of Geodesic Flows on Compact Surfaces of Genus g > 1 in the Class of Functions Analytic in Momenta... 

Theorem 5.4.3 (see [307]). The geodesic Sow in T M, where M is a two-dimensional Riemannian manifold, has an additional integral quadratic in momenta if and only if in any isothermic coordinates x, y the function X setting the metric satisSes equation (4), where R = R - 2iR2 is a holomorphic function of the variable z = X + ty, which is not identical zero and under transition to other isothermic coordinates is transformed in accordance with formula (7). 

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