Real part of a complex number

The set of real numbers The rt-dimensional vector space Real part of a complex number Sample-to-detector distance Guinier radius (i.e. radius of gyration)... 

The real part of a complex number is denoted by x = 8fz and the imaginary part by y = Sz. The complex conjugate z (written as z in some books) is the number obtained by changing i to —i ... 

Let us consider a complex number plane. On the abscissa we shall put the real part of a complex number and on the ordinates its imaginary part (Figure 2.6). Then any complex number can be written as... 

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. 

The sum 2fk,e> in eq. (10-10) corresponds to counting all the photons admitted by the photon-counting apparatus. Re (a) represents the real part of the (complex number) a. As expected, eq. (10-10) contains a direct decay term and interference terms. To study further the consequences of eq. (10-10) consider the expression for the differential counting rate ... 

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

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

We will assume in this chapter that the wave number k has a positive imaginary part, i.e. Rei/c < 0, here Re is the real part of the complex number, ik. Function A tends to zero as i —> 00, letting B = 0 and A = M/4tt, for this reason we obtain the known expression for the vector potential of a magnetic dipole in a uniform medium ... 

Re Reynolds number or real part of a complex function R(q) Fluctuation function f Reduced drop radius... 

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 ... 

Eq. (14.3) is the iterative equivalent of the TDSE. It does not contain the imaginary number i. If x, t) is just the real part of a complex wavefunction, then the equation will propagate forward, completely exactly, this real part of the complete wavefunction. The advantage of this is both one of storage and of computer time. Only half the computer storage is required for the real part of a wavefunction as compared for the complete complex wavefunction. The multiplication of two complex numbers requires four times as many operations as the multiplication of two real numbers. 

It is possible to represent complex numbers by means of a coordinate system. The real part of the complex number is designated along the x-axis, while the pure imaginary part is designated along they-axis, as shown in Fig. 1-8. Since x = rcos and y = r sin 0, any complex number can be written as... 

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 ... 

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. 

Complex plane plot — The complex number Z = Z + iZ", where i = v/-i, can be represented by a point in the Cartesian plane whose abscissa is the real part of Z and ordinate the imaginary part of Z. In this representation the abscissa is called the real axis (or the axis of reals) and the ordinate the imaginary axis (the axis of imaginaries), the plane OZ Z" itself being referred to as the complex plane [i]. The representing point of a complex number Z is referred to as the point Z. 

The real and imaginary parts of the complex numbers used here have no physical significance. This is simply a convenient way to represent the component vectors of stress and strain in a dynamic mechanical experiment. 

As is evident m the graphical representation of a complex number in Figure 1.1, two complex numbers are equal if and only if both the real and the imaginary parts are equal. Thus, an equation involving complex variables requires that two equations are satisfied, one involving the real terms, and one involving the imaginary terms. Commutative, associative, and distributive laws hold for complex... 

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