Complex Lorentz function

Thus, the transfer function is the complex Lorentz function, and the linear response in the time domain is the inverse Fourier transform of K (w) (cf. Section 4.1),... 

To facilitate the derivation of the dispersion relation, the discrete counterpart of the continuous permittivity, i.e., the numerical permittivity, is defined as the ratio of the discrete values of D and E. So, in analogy to the continuous case /xo num(complex permittivity function for the Lorentz medium is... 

Figure 10. The diffusive inodes of the periodic hard-disk Lorentz gas represented by their cumulative function depicted in the complex plane (ReF, ImF ) versus the wavenumber A .

In the 4D equation, space and time coordinates are inextricably entangled. Its mathematical solutions are hypercomplex functions, or quaternions, without a commutative algebra. Quaternions are used to describe what is known as spherical rotation, also called the spin function, and the complex rotation known as the Lorentz transformation of special relativity. 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

It is significant to note that the spin function also describes the complex rotation which defines the Lorentz transformation. An unexpected bonus of a fourdimensional quantum mechanics is this natural merger with relativity theory, the importance of which is beyond the scope of this work. Of more immediate relevance are the relativistic ramifications in chemical systems. 

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