Conjugate harmonic functions

The u x,y) of a differentiable function J z) is also called a harmonic function, whereas v(x,y) is called a conjugate harmonic function. 

Similar information to that obtained dining a conjugate gradient refinement is obtainable from second derivatives (curvature)1 R 51 93. For a harmonic function the gradient (linear matrix of first derivatives, [A]) multiplied by the curvature (Hessian matrix of second derivatives, [C]) should lead directly to the shifts (AX) to be applied in order to move toward the minimum (Eq. 3.3). 

The pioneers of quantum theory were unprepared for the discovery of two related, but unforeseen, new effects—the appearance of complex variables as an essential feature of the theory and pairs of conjugate variables that do not commute—a stark departure from classical mechanics. The appearance of complex variables is now readily appreciated, even in two-dimensional harmonic functions, defined by the equation ... 

So let us consider the -potential most simply as being replaced with such a Whittaker [1,56] decomposition. Then each of these scalar potentials—from which the A potential function is made—is decomposable into a set of harmonic phase conjugate wavepairs (of longitudinal EM waves). If one takes all the phase conjugate half-set, those phase conjugate waves are converging on... 

Suppose that Hamiltonian 7/(p, q) is expressed in a region around a saddle point of interest as an expansion in a small parameter e, so that the zero-order Hamiltonian Hq is regular in that region specifically, it is written as a sum of harmonic-oscillator Hamiltonians. Such a zero-order system is a function of action variables J of Hq only, and it does not depend on the conjugate angle variables 0. The higher-order terms of the Hamiltonian are expressed as sums of... 

As usual, there are the spherical harmonics or the complex conjugates (identified by the ). This expression can be inserted into the intermediate scattering function where one gets the following expression containing the regular Bessel functions ji ... 

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 ... 

The irreducible representations may be classified according to whether j is an integer or half of an odd integer. We shall consider here the former, which are the potentially real representations 22 b, p. 287). These representations in contrastandard form 4) can be transformed into real form by a constant unitary matrix, i.e. the same matrix for every element in the group. The elements of the constant matrix will be chosen such that the contrastandard, self-conjugate sets which form the bases for the potentially real representations in complex standard form are connected to the sets of the usual real spherical harmonics which form the bases for the real standard representations. From the constant matrix, the vector coupling coefficients pertaining to the real functions will be deduced. 

Elements of contrastandard, self-conjugate sets have been chosen in Ref. (4). The basis functions jm> are for integery represented by the complex standard spherical harmonics, which differ from the spherical harmonics Yjm= 0 jtn) 0 m) used by Condon and Shortly... 

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