Period functions

Each logarithm in the last temi can now be expanded and the (—n)th Fourier coefficient arising fi om each logarithm is — jn) zk-Y- To this must be added the n = 0 Fourier coefficient coming from the first, f-independent term and that arising from the expansion of second term as a periodic function, namely. 

IMPORTANT PROPERTIES WHICH SHOW A PERIODIC FUNCTION... 

Consider a periodic function x(t) that repeats between t = —r/2 and f = +r/2 (i.e. has period t). Even though x t) may not correspond to an analytical expression it can be written as the superposition of simple sine and cosine fimctions or Fourier series, Figure 1.13. 

I hus iq(x) is a periodic function which can be used to formulate acceptable wavefunctions ... 

Hence we see that this simple periodic function has just two terms in its Fourier series. In terms of the Sine and Cosine expansion, one finds for this same f(t)=Sin3t that an = 0, bn =... 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. 

To obtain the potential (jtaCr), we note that ( )2(r + mL) = ( )2(r) for any integer triple m, because this is simply a shifted sum over images. Thus ( )2(r) is a periodic function of r. Similarly, the source density p2, given by the sum of the co-ion densities over atoms... 

The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F t) with a period t, a Fourier series may be expressed as... 

D. 1. Menddeev enunciated his periodic law that the properties of the elements are a periodic function of their... 

Any periodic function can be represented as a series of sine functions having frequencies of co, 2co, 3co, etc. ... 

Mathematical theory shows that any periodic function of time, /(f), can be represented as a series of sine functions having frequencies a>, 2a>, 3ft), 4ft), etc. Function /(f) is represented by the following equation, which is referred to as a Fourier series ... 

The chemical and physical properties of dements are a periodic function of atomic number. 

Periodic function A physical or chemical property of elements that varies periodically with atomic number, 152 Periodic Table An arrangement of the elements in rows and columns according to atomic numbers such that elements with similar chemical properties foil in the same column,... 

The rows and columns of Mendeleev s table are meant to reflect the periodic function asserted to exist by this periodic law. 

It is no surprise that Mendeleev never gave precise mathematical expression to this periodic function . In fact, it would be impossible, we claim, to state at all precisely the content of Mendeleev s periodic law . (We are, of course, referring here to the law as articulated by Mendeleev himself and as understood by his contemporaries. There is no doubt that the subsequent development of chemistry has seen at least great progress toward the articulation of a precise version of the periodic law, based ultimately on quantum mechanics.24)... 

The applied field E is a periodic function of time t and has a frequency i o which is the frequency of laser emission. 

In order to clarify these ideas, let us calculate the power density spectrum of the real, periodic function given by... 

We shall now see how we can arrive at the same conclusion using the methods derived above, which, it should be emphasized, are applicable to a much larger class of functions than the periodic functions. 

If all xx,x2,- have been calculated and are periodic, F is then a known periodic function and if k is not an integer, there exists only one periodic solution of the form... 

When the DFT was employed to calculate the convolution, it means all signals and output were converted into periodic functions. This, more or less but inevitably, will bring about the periodic errors. 

Cartledge, G. H. (1928a). Studies on the periodic system. I. The ionic potential as a periodic function. Journal of the American Chemical Society, 50, 2855-63. 

There is an alternative - and for our purposes more powerful - way to estimate the discretization error, namely in terms of the Fourier expansion of a periodic -function. We write hf xk), see (A.l), as [24]... 

This analysis is based on the fact that any periodic function can be approximated, at any order, by a Fourier series ... 

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