It is easy to see

It is easy to see that the free constants U1, U2, … correspond to the potentials of a lesser degree, homogeneous in Euler\'s sense. These constants, except for the leading coefficient U0, can be set equal to zero without loss of generality. As a result, we can construct a set of linearly independent basis functions with consistently increasing polynomial degrees. Listed below are the expressions for symmetric and antisymmetric potentials homogeneous in Euler\'s sense which can serve as spectrographic media; notice that the degree of homogeneity k does not necessarily have to be an integer. The potentials with even degrees of y, symmetric with respect to z: The potentials with odd degrees of y, symmetric with respect to z: The potentials with even degrees of y, antisymmetric with respect to z: The potentials with odd degrees of y, antisymmetric with respect to z: Figs. 1–4 show equipotential surfaces of the fields from the list, represented by formulae (9)–(12). Symmetry or antisymmetry of a function with respect to the corresponding coordinate is fully equivalent to an expansion of the potential only in even or only in odd degrees of a coordinate. This is why expressions (9)–(12) in their ‘expanded’ form, when the y and z coordinates are interchanged, can be used in synthesizing particle optics systems of the required type. However, the OXY plane still remains the main plane where the principal motion of charged particles occurs. Detailed configurations of two-dimensional electrostatic and magnetostatic mirrors as applied to synthesizing opioid receptors spectrographs with ideal characteristics have been discussed, for example, in [4,7–10].
Conclusion We should note that the procedure for generating a quasi-polynomial is, in a sense, inverse to the process of expanding a harmonic function in a Taylor series in the vicinity of the plane of symmetry or antisymmetry by its given behavior along the plane of symmetry or antisymmetry [1,2]. So in this case, for the coefficients of the expansion in Taylor series, we start with the lowest polynomial coefficient and gradually move in a recurrent manner to the polynomial coefficients of all higher degrees. This procedure is, generally speaking, infinite, unless the behavior of the function along the plane of symmetry (or antisymmetry) is in itself a polynomial with respect to two variables. In contrast with this process, the construction of quasi-polynomials starts with the highest coefficient and gradually moves in a recurrent manner to polynomial coefficients of lower degrees, stopping at the lowest term in a finite number of steps. This procedure also differs from the method applied to using the generating function with respect to orthogonal polynomials of general form, for example, the Rodrigues formula for Hermite, Laguerre, Legendre, Chebyshev, Jacobi, Gegenbauer, Sonine, and other polynomials (see Ref. [16]).
Dedication
Introduction This article continues the investigations into spectrographic charged-particle optical structures which can serve as a basis for constructing effective devices with electric as well as magnetic fields. Potential structures of these fields ought to be homogeneous in Euler\'s sense [1,2]. According to our ideology, this property is very important, and a major condition for designing electric and magnetic spectrographs with high-performance characteristics such as resolution, sensitivity, transmission, energy dispersion and others, while the overall dimensions of the field-forming electrodes are rather small, and the device as a whole is compact. The property of homogeneity is expressed in the following analytical form. A continuous function of three variables, which is the potential U(x, y, z), is homogeneous in Euler\'s sense of k-th order, if the identity is fulfilled, where k is any real number. If the function U(x, y, z) is differentiable, then it can be described by a differential equation of first order