Holomorphic function

Theory of complex variables [1, 2] is used in order to connect current density and electric potential with one holomorphic function. The real part of it represents electric potential and the derivative is related with current density. The domain is divided into smaller subdomains where the holomorphic function is approximated with quadratic function and its derivative with piece wise linear function. These approximating functions obey continuity across subdomain interfaces. 

The fact that electric potential tp x, y) is harmonic, which comes from (2) allows to of use theory of complex variables, for 2D potential problems. Introducing harmonic conjugate function y/ x, y), where both of them satisfy Cauchy-Riemann equations (CR), one has holomorphic function ... 

Comparison of (3) and (5) shows that derivative of holomorphic function, which has real part equal to electric potential is related to the conjugated current density... 

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 ... 

This formula indicates that the reconstruction can be achieved but it will always have one undetermined parameter as a free parameter. In this particular case is the free parameter an additive constant in the imaginary part iy(x,y) of holomorphic function W(z). After performing derivative of H (z), the free parameter vanishes. It is therefore obvious that from electric potential one can recover the current density without any free parameter. In the case when data are known at discrete points, reconstruction, in general, is non-unique and can have any finite number of free parameters, which is evident from the following. Let an exact solution f(z) = u(x,y)+iv(x,y) of the problem be found by any method. It is obvious, that the modulus must be real therefore it satisfies the following ... 

In order to have sense the solution has to be sought in certain class of functions. Hereafter the solution is sought as a piecewise linear holomorphic functions. It will be shown that this set of functions allows reconstructing the current density with high accuracy. 

Similar mathematical formulation has been described in [3], where the required function was related to the modulus of holomorphic function. In this case is the required current density related directly with the derivative of holomorphic function W (z). 

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. 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

Conversely, assume T is a local isomorphism. Let Ox and l y denote the sheaves of holomorphic functions on X and Y. f induces, for all closed points iGX, all the following maps ... 

These conditions on the real and imaginary parts of a function w(x, y) must be fulfilled in order for u to be a function of the complex variable z. If, in addition, u and v have continuous partial derivatives with respect to x and y in some region, then w(z) in that region is an analytic function of z. In complex analysis, the term holomorphic function is often used to distinguish it from a real analytic function. 

The conformal transformation is an approach to calculate complex potential fields by means of transmapping two holomorphic functions such as z and w, as described earlier. The potential field on the z-plane can be obtained if a function w = f z) is known. When the potential field is not described by a known function, it can be analyzed by the Schwarz-Christoffel transformation, which is outlined here. 

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. 

PROOF Let t r — Af be an embedding of tori, t an induced mapping of universal coverings, and t C — C . Then the Jacobian matrix of t is a periodic holomorphic function on with a complete lattice of periods of rank 2p and is therefore constant. Consequently, t is an affine mapping, and we have come to the desired conclusion. 

Let an arbitrary holomorphic function R[z) be given in a certain coordinate neighbourhood of (x, y). Differentiating the first and second equations of the system (2) with respect to y and x corresponding and equating the right-hand sides of the equations derived, we obtain the following necessary condition for solvability of the system (2) with respect to the function 6 /y + h/5 = fc, + //15, which is equivalent to... 

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). 

Consider the solution of the basic equation (4) for an arbitrary holomorphic function R(z) 0. Go over to new coordinates, such that 5(ti ) = 1. According to formula (7) we then have w (z) = which given an explicit expression for... 

In addition, such a power series expansion is, however, only permitted for analytic, i.e., holomorphic functions and must never be extended beyond a singular point. Since the square root occurring in the relativistic energy-momentum relation Ep of Eq. (11.11) possesses branching points at X = p/nteC = i, any series expansion of Ep around the static nonrela-tivistic limit T = 0 is only related to the exact expression for Ep for non-ultrarelativistic values of the momentum, i.e., t < 1. This is most easily seen by rewriting Ep as... 

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