Since the real and imaginary parts of a complex number are independent of each other, a complex number is always specified in terms of two real numbers, like the coordinates of a point in a plane, or the two components of a two-dimensional vector. In an Argand diagram a complex number is represented as a point in the complex plane by a real and an imaginary axis. [Pg.8]

Im denoted the imaginary part of a complex quantity. ti,t4,t5,t6 are given in Table 3-1. [Pg.221]

The imaginary part of a complex number x+iy where x and y are real is y. incenter... [Pg.176]

Another limitation on acoustic properties is expressed by the Kramers-Kronig (KK) relations, which are general relations between the real and imaginary parts of a complex function. These relations were originally derived for optics but can be applied in many other areas as well. The essence of the relations is that the real and imaginary parts of the function are not independent of each other but one may be calculated from an integral of the other. As applied to complex modulus, the specific form of the relations is given elsewhere in this book (J. Jarzynski, A Review of the Mechanisms of Sound Attenuation in Materials). [Pg.43]

A wave can be characterized by an amplitude, frequency, and wavelength which may change with time or distance traveled from the source. We can express both the storage and loss properties of a sonic wave moving in a material concisely as the real and imaginary parts of a complex wavenumber k = co/c + ia, where c is the speed of sound, co is the angular frequency (=2 Jt/),/is frequency, / = V - 1, and a is the attenuation coefficient. Ultrasonic properties are often frequency dependent so it is necessary to define the wavelength at which k is reported. The dependency of k on frequency is the basis of ultrasonic spectroscopy. [Pg.133]

Recognize the real and imaginary parts of a complex number expressed in either cartesian or plane polar coordinates... [Pg.28]

The nomenclature of complex moduli and compliances is also often used. Here the out-of-phase component is made the imaginary part of a complex parameter thus the complex shear modulus G and the complex shear compliance J are defined as... [Pg.27]

In this equation, A9= 9- 9b is the relative incident angle. = rj )l nVc) is a scaling factor, where re = 2.818xlCf5 A is the classical electron radius and Vc is the volume of the unit cell (uc). (To separate the real and the imaginary parts of a complex quantity A, we use the notation A = A + iA", where A and A are real quantities.) From Equations (7), (9), and (10), it can be shown that the reflectivity approaches unity over a very small arc-second angular width a, defined as... [Pg.224]

Fourier analysis is very useful to describe the signals frequency content. However, it has a serious drawback. In a typical FT of a signal, it is impossible to tell when an event has happened because time information is lost. For all nonstationary signals that contain drift, trends, or abrupt changes, it is important to keep the time information in the transformation. In principle, time information can be found in the imaginary part of a complex FT. However, since only the real part is usually considered, the time information will not be available. [Pg.78]

The two signals (Eqs. (42) and (43)) are then considered as real and imaginary parts of a complex function, and a complex Fourier transform is performed in the computer. A further technical detail of the NMR spectrometer sketched in Fig. 13 is the so-called field-frequency lock. This implies that the rf-frequency is controlled in relation to the applied static magnetic field by monitoring the NMR signal of or F nuclei. [Pg.116]

Therefore, the real part of the complex solution to the complex form of Equation [IJ or [7] is the same as the real solution of the real form of Equation [1] or [7]. The two main advantages of complex notation are (a) simpler algebra in solving Equations [1] or [7], and (b) automatic separation of the in-phase and 90 -out-of-phase components (as the real and imaginary parts of a complex amplitude). [Pg.6]

It is therefore common to refw to x and x" as the mathematically "real" and "imaginary" parts of a "complex" quantity, X, even though x and x" clearly represent physically (and mathematically) real in-phase and 90 -out-of-phase amplitudes of a real displacement. [Pg.6]

imaginary parts of a complex munber. For example, the following statement assigns the imaginary part of a, which is a Complex, to X, which is a double number. [Pg.33]

In the mathematical data treatment, it is cmivenient to use a complex amplitude and Euler equations to represent the in-phase and out-of-phase signals as real and imaginary parts of a complex signal ... [Pg.291]

The procedure described above is not really a CNLS approach unless/ and/ are the real and imaginary parts of a complex variable. But as we have seen, Z and 6 are not, although ln Z and 9 and Z and Z" are. Since we sometimes are inter-... [Pg.196]

The parameters of the Hamiltonian (3), i.e. the frequencies and the number of oscillators (more specifically, the strength of oscillators) are determined by the imaginary part of a complex dielectric function a(k, co) which characterizes the dielectric losses for polarization fluctuations in a medium. This model is, strictly speaking, applicable to homogeneous isotropic media in which the spatial correlations of polarization fluctuations SP r)dP r ), which determine the dependence of s k, co) on the wave vector k, depend on the difference of coordinates r—r only. [Pg.15]

The Kronig-Kramers relationships are a very general set of integral transforms that find wide application in phjreical problems. They are intimately related to Hilbert transforms which, subject to certain integrability and analyticity conditions, allow the real and imaginary parts of a complex function f(z) = u iv to >t expressed as a pair of transform mates. This property follows from the fact that u and v are not completely independent when / z) is analytic in the whole upper half of the complex plane. [Pg.87]

The imaginary part of any complex number is a real number multiplied by i = V. (The symbol = is used throughout this text to indicate a definition, as opposed to the = symbol, used for equalities that can be proved mathematically.) This relationship between i and — 1 allows the imaginary part of a complex number to influence the real-number results of an algebraic operation. For example, if a and b are both real numbers, then a + ib is complex, with a the real part and ib the imaginary part. The complex conjugate of a -F ib, written (fl + ib), is equal to a — ib, and the product of any number with its complex conjugate is a real number ... [Pg.8]

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