Complex exponentials

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

The diffracted amphtude from illuminating such a grating with a unit plane wave normal to the surface is easily calculated again by resolving equation 9 into complex exponentials (as in eq. 10) where is the mUi Bessel function. 

Normally the orbitals are real, and the unitary transformation becomes an orthogonal transformation. In the case of only two orbitals, the X matrix contains the rotation angle a, and the U matrix describes a 2 by 2 rotation. The connection between X and U is illustrated in Chapter 13 (Figure 13.2) and involves diagonalization of X (to give eigenvalues of ia), exponentiation (to give complex exponentials which may be witten as cos a i sin a), follow by backtransformation. 

The Fourier transform H(f) of the impulse response h(t) is called the system function. The system function relates the Fourier transforms of the input and output time functions by means of the extremely simple Eq. (3-298), which states that the action of the filter is to modify that part of the input consisting of a complex exponential at frequency / by multiplying its amplitude (magnitude) by i7(/)j and adding arg [ (/)] to its phase angle (argument). 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

In the case of the reciprocal sum, two methods have been implemented, smooth particle mesh Ewald (SPME) [65] and fast Fourier Poisson (FFP) [66], SPME is based on the realization that the complex exponential in the structure factors can be approximated by a well behaved function with continuous derivatives. For example, in the case of Hermite charge distributions, the structure factor can be approximated by... 

It is noted that the solutions are real rather than complex exponentials. Since the constant D = 0 the probability density remains finite as x —> oo. In the region x < 0, as before,... 

Let us consider the Fourier representation of f(t) in terms of the complex exponentials introduced above. For an arbitrary periodic function f(t), we can write... 

By using complex exponential functions instead of trigonometric functions, we only have... 

The complex exponential separates into factors involving only a single variable. Lumping the constant S with the mask, we now have for the correction term ... 

If the specimen is moved away from the focal position, then this will cause a phase shift that depends on 6. If the wavenumber in the coupling fluid is k = 2n/Xo, then the z component of the wavevector is kz = k cos 6. Defocusing the specimen by an amount z causes a phase delay of 2zkz, or 2zk cos 0 (the factor of two arises because both the incident wave and the reflected wave suffer a change in path length). Expressing this phase delay as the complex exponential of a phase angle, the response of the microscope with a defocus z is... 

The asterisk designates the complex conjugate. Moreover, we note that the above Eqs. 2.46 and 2.47 imply positive as well as negative frequencies. In some physics applications, an appearance of negative frequencies may be confusing only positive frequencies may have physical meaning. In such cases one may rewrite the above inverse tranform in terms of positive frequencies, using a well-known relationship between the complex exponential function and the sine and cosine functions. 

Now, it is clear that since t > 0 in the complex exponential in the integrand of eq. (12-29) we must close the contour in the lower half plane. Thus we analytically continue the functions Rij(E) from the first to the second Riemann sheet through the real T -axis (the branch cut of G(E)), i.e., we set... 

All of the usual properties of exponentials (Equations 1.13 and 1.14) also apply to complex exponentials. For example, the product of two exponentials is found by sum-... 

The solution to this equation is a complex exponential function. Hence... 

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

The solution is a complex exponential, subject to considerable difficulty in numerical calculation. However, it is appropriate to evaluate Equation 14 graphically by calculating vt from Equation 1 for values of Dt and t obtained by solving Equation 12 at selected values of D0. This integration gives the distance the drop has fallen, h, in time t for stated values of D0 and Ap/P. 

Table 4.1 Four-digit numerical values for all the input spectral parameters the real Re(v/,) and imaginary lm(v/<) parts of the complex frequencies Vk, and the absolute values dk of the complex amplitudes dk of 25 damped complex exponentials from the synthesized time signal Eq. (42) similar to a short echo time ( 20 ms) encoded FID via MRS at the magnetic field strength 80 = 1.5T from a healthy human brain [71]. Every phase 4>k of the amplitudes is set to zero (0.000), for example, each dk is chosen as purely real, dk= dk exp (Uj>k) = dk. The letter M denotes the feth metabolite...

The term Fourier transform usually refers to the continuous integration of any square-integrable function to re-express the function as a sum of complex exponentials. Due to the different types of functions to be transformed, many variations of this transform exist. Accordingly, Fourier transforms have scientific applications in many areas, including physics, chemical analysis, signal processing, and statistics. The continuous-time Fourier transforms are defined as follows [1-3] ... 

It is often convenient to use a complex number representation for sound waves [2]. The harmonic wave is represented by the complex exponential. 

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 ... 

Equilibrium centrifugation can be used to analyse the behaviour of mixtures and interacting systems. However, it should be noted that this technique does not separate individual spedes as do sedimentation velodty experiments, and analysis of the complex exponential distributions of material which occur when mixtures are centrifuged is only feasible when the number of components is very limited. 

As you can see, the group exhibits separably degenerate representations and it is customary to specify these explicitly, so that complex exponentials appear in the character tables of such groups. 

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