Complex exponential form

The complex exponential form derives from de Moivre s relation cos 0 + isin< = exp ( < ). 

We return now to the problem of adding the scattered waves from each of the atoms in the unit cell. The amplitude of each wave is given by the appropriate value of/for the scattering atom considered and the value of (sin 0)/X involved in the reflection. The phase of each wave is given by Eq. (4-4) in terms of the hkl reflection considered and the iww coordinates of the atom. Using our previous relations, we can then express any scattered wave in the complex exponential form... 

The generalized scattering equation can be expressed by a complex exponential form... 

Solving Eq. (4.48) and its complex-conjugate equation = cosO — i sin0 for sin0 and cos0, we can represent these trigonometric functions in complex exponential form ... 

Converting the cosine/sine form to the complex exponential form allows many manipulations that would be very difficult otherwise (for an example, see Section 2 of Appendix A Convolution and DFT Properties). But, if you re totally uncomfortable with complex numbers and Euler s Identity (or with the identities of IS"" century mathematicians in general), then you can write the DFT in real number terms as a form of the Fourier Series ... 

Integrals involving trigonometric functions can often be evaluated using the identities of Problem 1.26. Use the complex-exponential form of the sine function to verify Eq. (2.27) for the particle-in-a-box wave functions. 

While the complex exponential form has an imaginary part.jv/w (tor) to the waveform, in physical systems such as speech, this does not exist and the signal is fully described by the real part. When requiring a physical interpretation, we simply ignore the imaginary part of... 

Using the complex exponential form, we can write a more general form of the Fourier synthesis equation ... 

The discrete-time Fourier series (DTPS) for discrete-time waveforms x n) of period N can also be given in three forms however, the complex exponential form is by far the most common. 

If we now include the phase and scattering factor expressions we used in Eqs. 11 and 12, we will obtain an equation that fully describes the scattering of the wave from a lattice atom, in complex exponential form (Eq. 15) ... 

Ligand release in the reaction of bis(A/-alkylsalicylaldiminato)zinc(II) complexes with ammonium ions in acetonitrile follows first-order kinetics, albeit in double exponential form for the t-butyl complex. The first bond to break is Zn-0 (317). 

In order to describe the material properties as a function of frequency for a body that behaves as a Maxwell model we need to use the constitutive equation. This is given in Equation (4.8), which describes the relationship between the stress and the strain. It is most convenient to express the applied sinusoidal wave in the exponential form of complex number notation ... 

D = D° exp(-ac ), where D is the diffusion, D represents the zero-concentration limit, c is the concentration, a and v are parameters, fits the data from a wide variety of probes and matrix polymers ( ). Several theoretical justifications for this behavior have been presented (97-1011. but it is not possible to tell yet which, if any, is uniquely correct. The treatments range from simple physical considerations (98) to treatments of hydrodynsumical interaction of probe and matrix (97,991. Other more complex and general treatments (1001 do not explicitly arrive at the stretched exponential form, but do closely fit the available data. Much more work needs to be done on probe diffusion in such transient networks. Beyond enhancing the arsenal of gel characterization, the problem is quite fundamental to a number of other important processes. 

Rather than tackle the full problem immediately, it will help to start with some simplification of the dimensionless equations and then build up to the full complexity in stages. At the first level of approximation we will make use of the typically small value of y and replace the full Arrhenius dependence by the exponential form ea. We also begin with those systems for which the inflow and ambient temperatures are the same, so 9C = 0. 

Entropy is an equilibrium thermodynamic entity that is interpreted mechanically as the degree of disorder in a system. From Eq. (8.4), it is therefore seen that (1) the preexponential factor A [Eq. (8.1) can be interpreted as being related to the organization of a reactant in an enzyme as the transition-state complex is formed and (2) the exponential factor relates to the enthalpy (heat) of the reaction. 

Normally, for semiconductors, Csc < CH so CT Csc. Roat may be varied systematically and the decay of j can often be approximated by a single exponential form, i.e. kr 1/RtCi. or kT potentials well positive of V, the long-time transient time-constant t (Rm + Rout)Csc, and a plot of x vs. R]oad (sflin + Rout) is linear, as shown in Fig. 105. Confirmation of this is obtained from the fact that 1/t2 obeys the Mott-Schottky relationship. At potentials close to V, kec becomes much larger and the decay law more complex. 

An acoustic wave is a traveling periodic pressure disturbance. This wave travels at a speed c dependent on the properties of the medium and the type of motion associated with the wave. The periodic nature of the acoustic wave is (for present purposes) taken to be a sinusoidal oscillation occurring at a frequency f. At any location x and instant in time t, the pressure associated with this traveling wave can be expressed as a cosine wave, or in a mathematically equivalent form as the real part of a complex exponential ... 

Now, according to the transition-state theory of chemical reaction rates, the pre-exponential factors are related to the entropy of activation, A5 , of the particular reaction [A = kT ere k and h are the Boltzmann and Planck constants, respectively, and An is the change in the number of molecules when the transition state complex is formed.] Entropies of polymerization are usually negative, since there is a net decrease in disorder when the discrete radical and monomer combine. The range of values for vinyl monomers of major interest in connection with free radical copolymerization is not large (about —100 to —150 JK mol ) and it is not unreasonable to suppose, therefore, that the A values in Eq. (7-73) will be approximately equal. It follows then that... 

Although these reactions are complex at a biochemical level, their kinetics approximate to reactions of the first order. Thus, the kinetics of inactivation of populations of pure cultures of micro-organisms take the typical exponential form of reactions of the first order. What this means in experimental practice is that there is a linear relationship when numbers of microorganisms held at high temperatures are plotted on a logarithmic scale against time plotted on an arithmetic scale (Fig. 1). 

By Fourier transformation, a signal is decomposed into its sine and cosine components [Angl]. In this way, it is analysed in terms of the amplitude and the phase of harmonic waves. Sine and cosine functions are conveniently combined to form a complex exponential, coscot 4- i sinwt = exp icomplex amplitudes of these exponentials constitute the spectrum F((o) of the signal f(t), where co = In IT is the frequency in units of 2n of an oscillation with time period T. The Fourier transformation and its inverse are defined as... 

Fourier analysis of descriptors is performed by the DFT, which is the sum of the descriptor g r) over all distances r multiplied by a complex exponential. With a descriptor consisting of n components expressed in its discrete form g[x] x is the index of a discrete component), the DFT can be written as... 

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