Equating imaginary parts

It is possible to understand the fine structure in the reflectivity spectrum by examining the contributions to the imaginary part of the dielectric fiinction. If one considers transitions from two bands (v c), equation A1.3.87 can be written as... 

The observable NMR signal is the imaginary part of the sum of the two steady-state magnetizations, and Mg. The steady state implies that the time derivatives are zero and a little fiirther calculation (and neglect of T2 tenns) gives the NMR spectrum of an exchanging system as equation (B2.4.6)). 

As was mentioned above, the observed signal is the imaginary part of the sum of and Mg, so equation (B2.4.17)) predicts that the observed signal will be tire sum of two exponentials, evolving at the complex frequencies and X2- This is the free induction decay (FID). In the limit of no exchange, the two frequencies are simply io3 and ici3g, as expected. When Ids non-zero, the situation is more complex. 

In this equation, the index j runs over all the transitions and the exponents have both real and imaginary parts, which... 

In this equation, the primes on the imaginary parts indicate that the Lamior frequencies and coupling constants will be different. Also, if the equilibrium constant for the exchange is not 1, then the forward and reverse rates will not be equal. Note that the 1,2 block, in the top right, represents the rate from site 2 into site 1. 

Real and imaginary parts of this yield the basic equations for the functions appearing in Eqs. (9) and (10). (The choice of the upper sign in these equations will be justified in a later subsection for the ground-state component in several physical situations. In some other circumstances, such as for excited states in certain systems, the lower sign can be appropriate.)... 

For our purposes we can neglect the imaginary part of Equation (3.16), i sin27ivt, and then it is apparent that f t) is a sum of cosine waves, as we had originally supposed. Fourier transformation allows us to go from f t) to F(v) by the relationship... 

Principles in Processing Materials. In most practical apphcations of microwave power, the material to be processed is adequately specified in terms of its dielectric permittivity and conductivity. The permittivity is generally taken as complex to reflect loss mechanisms of the dielectric polarization process the conductivity may be specified separately to designate free carriers. Eor simplicity, it is common to lump ah. loss or absorption processes under one constitutive parameter (20) which can be alternatively labeled a conductivity, <7, or an imaginary part of the complex dielectric constant, S, as expressed in the foUowing equations for complex permittivity ... 

Ma is defined by Equation 5.1.8 and / is defined by Equation 5.1.7. Typical curves for fhe imaginary part of the transfer frmction, lm[Tr], are plotted in Figure 5.1.13. These curves are calculated for a flame speed of 0.3 m/s, the other parameters in the coefficients A, B, C, and D are appropriate for a lean mefhane flame. The response is shown for three typical dimensionless wave numbers, kS = 0.01,0.03, and 0.1, which correspond to dimensional... 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

The frequency-domain spectrum is computed by Fourier transformation of the FIDs. Real and imaginary components v(co) and ifi ct>) of the NMR spectrum are obtained as a result. Magnitude-mode or powermode spectra P o)) can be computed from the real and imaginary parts of the spectrum through application of the following equation ... 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

Taking the real and imaginary parts of the left- and right-hand sides of the newly rewritten observational equation, one obtains... 

Interaction ofthe electrons in the framework of the self-consistent field approximation is accounted for by considering the induced density fluctuations as a response of independent particles to Oext + Poissons equation [2], This means, physically, that collective excitations of the electrons can occur, taken into account via a chain of electron-holeexcitations. These collective excitations show up in S(q, ) as a distinct energy loss feature. Figure 2 shows the shape of the real and imaginary parts of the dielectric function in RPA (er(q, ), Si(q, )) and the resulting dielectric response... 

From the imaginary part equation, the ultimate frequency is u = Vl 1. Substituting this value in the real part equation leads to the ultimate gain Kc u = 60, which is consistent with the result of the Routh criterion. 

We have two equations after collecting all the real and imaginary parts, and requiring both to be... 

Thus we have either = 0 or - 2 + (1 + Kc) = 0. Substitution of the real part equation into the nontrivial imaginary part equation leads to... 

Collecting terms of the real and imaginary parts provides the two equations ... 

The absorption signal, of course, is the imaginary part of eqn (5.18) the equation is too horrible to contemplate, but computer-simulations, such as those shown in Figures 5.3 and 5.4, are relatively easy to produce. There are two limiting cases where the equations are easier to understand. In the slow exchange limit, where xA-1 and xB 1 are both small compared with ooA - ooBl, the... 

Harmonic Functions Both the real and the imaginary parts of any analytic function f = u + iv satisfy Laplace s equation second partial derivatives and satisfies Laplace s equation is called a harmonic Function. 

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