Complex number imaginary part

Complex—Value stored as two words, one representing the real part of the number and the other representing the imaginary part. 

AIMAG Returns the imaginary part of a complex number... 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

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 ... 

We can plot the real and imaginary parts of G(jco) on the v-planc with co as the parameter—the so-called Nyquist plot. Since a complex number can be put in polar coordinates, the Nyquist plot is also referred to as the polar plot. 

It should be recognized that the discrete Fourier coefficients G(x, y, co) are represented by complex numbers. The real part Re(G(x, y, to)) of the complex number represents the amplitude of the cosine part of the sinusoidal function and the imaginary part Im(G(x, y, co)) represents the amplitude of the sine wave. 

Presented in this manner, the analysis may proceed similarly to the treatment obtained from the Fourier analysis. C is the zero frequency component of the fit and A and B may be treated as the real and imaginary parts of the complex number. 

A complex number consists of two parts a real and a so-called imaginary part, c = a + ib. The imaginary part always contains the quantity i, which represents the square root of -1, i = /—1- The real and imaginary parts of c are often denoted by a = R(c) and b = 1(c). All the common rules of ordinary arithmetic apply to complex numbers, which in addition allow extraction of the square root of any negative number. If... 

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. 

Complex a(s) may be interpreted as representing two mixtures simultaneously, one corresponding to the real, one to the imaginary part. This will not cause any inconsistency with the operations we shall be performing with the ensemble operators. Alternatively, we could simply restrict ourselves to real a(s), as the real numbers suffice for the full reduction of representations of [Pg.48]

The roots are now complex numbers with real and imaginary parts. 

Gfj , is a complex number, so it can be represented in terms of a real part and an imaginary part ... 

Knowing the complex number C, its real and imaginary parts can be found by using the statements... 

This equation is exact and constitutes an iterative equation equivalent to the time-dependent Schrodinger equation [185,186]. The iterative process itself does not involve the imaginary number i therefore, if h(f) and )( — x) were the real parts of the wavepacket, then (f + x) would also be real and would be the real part of the exact wavepacket at time (f + x). Thus, if )( ) is complex, we can use Eq. (4.68) to propagate the real part of 4>(f) forward in time without reference to the imaginary part. 

Some of you may be familiar with the concept of complex numbers that have a real and imaginary part of the form a + where i = (If you ve never heard... 

Quaternions are similar to complex numbers but of the form a + hi + cj.+ dk with one real and three imaginary parts. The addition of these 4-D numbers is fairly easy, but the multiplication is more complicated. How could such numbers have practical application It turns out that quaternions can be used to describe the orbits of pairs of pendulums and to specify rotations in computer graphics. 

In conducting media, the wave number becomes complex [5], and by separating real and imaginary parts, we can obtain the Beltrami equations ... 

The only potential trouble is A = , which gives the return value of the status flag ER = 1. The return value of MR is the number of real roots. If NR = 3, the real roots will occupy the variables XI, X2 and X3. If MR = 1 then the only real root will occupy X, whereas you will find the real and imaginary parts of the conjugate complex pair in the variables XR and XI, respectively. 

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