Representation in the complex plane

Given the different phase angles of resistances and reactances described above, representation in two dimensions is useful. On the x-axis the 

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Figure 2.4 Integration contour for the Cauchy representation in the complex plane of a function of Nevanlinna type showing the deformation around the bound states and the cut along the real axis.

A2.4 Representation in the complex plane A2.5 Resistance and capacitance in series A2.6 Resistance and capacitance in parallel A2.7 Impedances in series and in parallel A2.8 Admittance... 

Fig. A2.1. Representation in the complex plane of an impedance containing resistive and capacitive components.

For such an iteration function scheme, a fractal attractor exists. The best known example for obtaining a fractal set is the square representation in the complex plane... 

For a long time it has been the only way, by the use of the Laplace method. Such a method is based on the representation in the complex plane of the confluent hypergeometric functions and the use the residues theorem. But it needs the approximation Z2a2 C ft2, which allows one to replace in (12.16)... 

An analytical integration of an integrodifferential equation under a singular time boundary is always a complicated matter. The treatment of the method, based on a representation of the delta functional as a Fourier transform, and working in the complex plane, would be out of place in this report. It can be found in detail in Ref. 7) where also the solution obtained is discussed. It is shown that this solution is especially simple if the elution curves show a positive skewness, i.e. if they are tailed on the right-hand-side of their maximum (this is always true in PDC and GPC). A renormalization of the found concentration profile and a recalculation of the coordinates (z, t) to the elution volumina (V, V) then yield the spreading function of the considered column (Greschner 7))... 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

The representation of structure factors as vectors in the complex plane (Qr complex vectors) is useful in several ways. Because the diffractive contributions of atoms or volume elements to a single reflection are additive, each contribution can be represented as a complex vector, and the resulting structure factor is the vector sum of all contributions. For example, in Fig. 6.4, F represents a structure factor of a three-atom structure, in which f), f2, and f3 are the atomic structure factors. 

Figure 12.1. Argand diagram for the representation of complex numbers in the complex plane C.

FIGURE 6.1 Left representation of z = x + iy in the complex plane, showing the magnitude z and phase 0. Right the quantity e ° is always on the unit circle, and is counterclockwise hy an angle 9 from the x-axis. 

According to the above calculations, a graphical representation of the AC impedance of a series RC circuit is presented in Figure 2.19. As shown in the complex plane of Figure 2.19, the AC impedance of a series RC circuit is a straight vertical line in the fourth quadrant with a constant Z value of R. 

Fig. IIL Vector representation of the impedance Z(co) in the complex plane. ReZ and ImZ are the real and the imaginary components of the impedance, respectively.

Figure 2.12. Nodal properties of the transition densities of the first four transitions in benzene, a) Representation of the complex LCAO coefficients of HOMOs 01 and 0, as well as LUMOs nd by means of a phase polygon. Each coefficient has the absolute magnitude n and the complex phase shown by a dot in the complex plane of which the real and imaginary axes are abscissa and ordinate, b) Representation of the overlap densities evaluated from the complex coefficients, and c) values of the overlap densities at the vertices of the perimeter and the resulting nodal properties.

In this polar representation, r is called the modulus or magnitude of z (r = z ), and the angle 0 is called the argument or phase of z. Note that as 0 varies from 0 to 2tz, el scribes a circle of unit radius in the complex plane (see Figure A.2), while its real part oscillates as cos 0. 

FIGURE 3.5 Geometric representation of multiplication by i. The point iz is obtained by 90° counterclockwise rotation of z in the complex plane. 

Some very elegant representations of special functions are possible with use of contour integrals in the complex plane. 

A convenient representation of the complex number z is as a point in the complex plane (Fig. 1.3), where the real part of z is plotted on the horizontal axis and the imaginary part on the vertical axis. This diagram immediately suggests defining two... 

We will use, during the following applications, a Cartesian representation of Z in the complex plane (real part as x-coordinate, imaginaiy part as y-coordinate). This representation is also called the Nyquist representation. 

In the Cartesian representation of the complex plane, this eqnation is represented by a R/2 radius semicircle whose center is on the x-axis at (r + y). 

Figure 4.11 Representation of an impedance value Z(cu) in the complex plane.

While the FT of the even cosine function is real, the result for sin(27rvot) is imaginary (S(v — vo)- --b Vo))/(2i). Since the sine function is 90° out of phase to the corresponding cosine, it is clear that the imaginary axis is used to keep track of phase shifts, consistent with the polar representation of a complex number x + iy = re with r = Jx - -y and 4> = tan (y/x). In this phasor picture, positive and negative frequencies can be interpreted as clockwise and counterclockwise rotations in the complex plane. 

FIGURE 24.22. Graphical representation of the storage and loss moduli and E" as components of a vector Ecomplex plane. d is the absolute value of the dynamic modulus. The corresponding compliances are shown in the right hand part of the figure. 

In the complex-plane representation of the impedance behavior of a parallel RC circuit, it is convenient to identify the maximum (so-called top point ) in the semicircular plot which is at a critical frequency O) = l/RC, the reciprocal of the time constant for the response of the circnit. A similar sitnation arises for a series RC... 

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