Numbers complex

Complex numbers are numbers of the form a + iA, where a and A are real numbers and i is the unit imaginary number with the property i2 = — 1. The ordinary operations of the algebra of real numbers can be performed in exactly the same way with complex numbers by using the multiplication table for the complex number units l,i shown in Table 12.1. Thus, the multiplication of two complex numbers yields 

Complex numbers may be represented by ordered number pairs [a, A] by defining [a, 0] = a[l, 0] to be the real number a and [0, 1] to be the pure imaginary i. Then 

Exercise 12.1-1 Prove the associative property of the multiplication of complex numbers. 

The distance from the origin 0 to the point Pisr (a2 + A1)A, which is called the norm or modulus of the complex number [a, A], The product of two complex numbers is as follows  

Since the exponential function may be defined everywhere in the complex plane, we may expand exp(i0) and, using the series expansions for the trigonometric functions, obtain Euler s formula 

Imaginary numbers have been defined into existence by mathematicians. They cannot be used to represent any physically measured quantity, but turn out to be useful in quantum mechanics. The imaginary unit is called i (not to be confused with the unit vector i) and is defined to be the square root of — 1  

If is a real number, the quantity i b is said to be pure imaginary, and if a is also real, the quantity 

The real number b is called the imaginary part of c and is denoted by 

All of the rules of ordinary arithmetic apply with complex numbers. The sum of two complex numbers is obtained by adding the two real parts together and adding the two imaginary parts together. If ci = ai + rfciand C2 = U2 + ib2, then 

The product of two complex numbers is obtained by the same procedure as multiplying two real binomials. 

Most of the numbers encountered in chemistry are known as real numbers. Examples are 

Note that the integers are a subset of the set of real numbers. 

The concept of imaginary numbers is rather less common. Consider the equation 

Subtracting 1 from each side of this gives 

Taking the square root of either side then gives 

The use of complex numbers is not obligatory, but they greatly simplify mathematical operations. 

Let us consider first a vector R rotating with a constant angular frequency CO = 2nf, where/is the frequency in s or Hz and co is in radians s (Fig. 2.6). 

The projection of R on the x- and y-axes, R and Ry, can be calculated using simple trigonometry  

In complex analysis, a projection on the x-axis is called the real part of vector / , and a projection on the y-axis is called the imaginary part. This is a simple way of distinguishing between these two projections, but, as we will see below, it simplifies considerably the calculations. The angle q can be obtained as 

The imaginary unit is defined as/ = — 1, and the complex plane has two axes, x, which is real, and y, which is imaginary all the real numbers on the y-axis are multiplied by the imaginary rmit j. This means that any point on the complex plane has two parts a real part on the x-axis and an imaginary part on the y-axis. This is illustrated in Fig. 2.7. Vector R may be written as 

We have seen that the wave function can be complex, so we now review some properties of complex numbers. 

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 

If Z is a real number, its imaginary part is zero. Thus z is real if and only if z — z -Taking the complex conjugate twice, we get z back again, (z ) = z. Forming the product of z and its complex conjugate and using i = -1, we have 

It is easy to prove, either directly from the definition of complex conjugate or from (1.31), that 

For the absolute values of products and quotients, it follows from (1.31) that 

The complex number z can be represented 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 quantities that characterize the complex number z the distance r of the point z from the origin is called the absolute value or modulus of z and is denoted by z the angle 6 that the radius vector to the point z makes with the positive horizontal axis is called the phase or argument of z. We have 

It is frequently helpful to deal with a number 2 in the complex plane, where the x axis represents the real part of the number and the y axis represents the imaginary part of the number, designated by the coefficient i = V—1. A point in the complex plane may be represented by a pair of numbers, x and y, or by absolute value (modulus) r and an angle 6. The quantities are related as follows  

The relations in Eq. C.2 may be added or subtracted to generate expressions for sin 0 and cos 6. Also, from Eq. C.2, we have the following relations  

In addition to vectors in the space domain, vectors may also be defined in the time domain, in particular rotating vectors with a fixed initial point at the origin of a Cartesian coordinate system. A variable such as the electric field strength may be a vector both in time and space. Every space vector may also be a time vector, and often it is not clear what sort of vector an author actually is dealing with. Vectors in the time domain are used for sine waves when the maxima do not occur simultaneously. These two-dimensional (planar) time vectors are more conveniently represented by complex numbers. 

A complex number is an ordered pair of real numbers, for instance G and B. Introducing the imaginary unit j = the complex number Y = G + jB. G is the real part and can be written Y, and B the imaginary part written Y . Y or Y is called the absolute value, magnitude, or modulus, and the phase angle is cp = arctan B/G. 

A real number G can be regarded as a position on a number line. A complex number Y can be regarded as a point in the plane of a special Cartesian coordinate system the complex plane, also called the Argand or Wessel diagram. G is an ordinary real number situated on the real x-axis. j (actually j) indicates that B is to be situated on the imaginary y-axis. B is a real number, jB is an imaginary number, Y is a complex number. 

Y = G — JB is ealled the eomplex conjugate to Y. Often the complex conjugate is used to obtain positive values for the imaginary component in the Wessel diagram. Impedance loci for instance are usually plotted with the circular arcs up, so instead of Z = R + jX, 

Summing up What is the difference between a space vector and a complex number in our context  

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The two factors on the right are both positive, real numbers less than one. If the magnitudes of U(h and U h ) are both close to one, therefore, the magnitude of the difference between the temis within the brackets on the left (complex numbers in general) must be small. 

Since this is a complex number, it can be separated into an amplitude and a phase and written as ... 

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

Thus D is a diagonal matrix that contains diagonal complex numbers whose nonn is 1. By recalling Eq. (57), we get... 

As the D matrix is a diagonal matrix with a complex number of norm exponent of Eq. (65) has to fulfill the following quantization mle ... 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

Making use of the polar representation of a complex number, the nuclear wave function can be written as a product of a real amplitude, A, and a real phase, S,... 

Thus B is a diagonal mati ix that contains in its diagonal (complex) numbers whose norm is 1 (this derivation holds as long as the adiabatic potentials are nondegenerate along the path T). From Eq. (31), we obtain that the B-matrix hansfomis the A-matrix from its initial value to its final value while tracing a closed contour ... 

The IE scheme is nonconservative, with the damping both frequency and timestep dependent [42, 43]. However, IE is unconditionally stable or A-stable, i.e., the stability domain of the model problem y t) = qy t), where q is a complex number (exact solution y t) = exp(gt)), is the set of all qAt satisfying Re (qAt) < 0, or the left-half of the complex plane. The discussion of IE here is only for future reference, since the application of the scheme is faulty for biomolecules. 

Fiv. 110 The A rgani diagram used to represent complex numbers. 

Arithmetical operations on complex numbers are performed much as for vectors. Thus, if a j hi and y = c + di, then ... 

A lommonly used relationship involving complex numbers is ... 

XI. Complex Numbers, Fourier Series, Fourier Transforms, Basis Sets... 

Complex numbers ean be thought of as points in a plane where x and y are the abseissa and ordinate, respeetively. This point of view prompts us to introduee polar... 

Complex numbers ean be added, subtraeted, multiplied and divided like real numbers. For example, the multiplieation of z by z gives ... 

A very useful way to simplify Eq. (10.65) involves the complex number e in which i = / 1 equals cos y + i sin y. Therefore cos y is given by the real part of e y. Since exponential numbers are easy to manipulate, we can gain useful insight into the nature of the cosine term in Eq. (10.65) by working with this identity. Remembering that only the real part of the expression concerns us, we can write Eq. (10.65) as... 

It is the net intensity, not the electric field, which concerns us. We previously used the fact that intensity is proportional to E to evaluate i. Using complex numbers to represent E requires one slight modification of this procedure. In the present case we must multiply E by its complex conjugate -obtained by replacing / 1 by to evaluate intensity ... 

Zinc arc spraying is an inexpensive process in terms of equipment and raw materials. Only 55—110 g/m is required for a standard 0.05—0.10 mm Zn thickness. It is more labor intensive, however. Grit blasting is a slow process, at a rate of 4.5 m /h. AppHcation of an adhesive paint layer is much quicker, 24 m /h, although the painted part must be baked or allowed to air dry. Arc sprayed 2inc is appHed at a rate of 9—36 m /h to maintain the plastic temperature below 65°C. The actual price of the product depends on part complexity, number of parts, and part size. A typical price in 1994 was in the range of 10—32/m. ... 

In the real-number system a greater than h a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow bl > x bl > y z Z9 z- + bgl bi - Zol Ibil — zo z- > (bl -I- lyl)/V2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities. 

Sufficient Conditions for the Existence of Laplace Transform Suppose/ is a function which is (1) piecewise continuous on eveiy finite intei val 0 < t exponential growth at infinity, and (3) Jo l/t)l dt exist (finite) for every finite 6 > 0. Then the Laplace transform of/exists for all complex numbers. s with sufficiently large real part. 

To determine the pipeline potentials, the resultant induced field strengths have to be included in the equations in Section 23.3.2. Such calculations can be carried out with computers that allow detailed subdivision of the sections subject to interference. A high degree of accuracy is thus achieved because in the calculation with complex numbers, the phase angle will be exactly allowed for. Such calculations usually lead to lower field strengths than simplified calculations. Computer programs for these calculations are to be found in Ref. 16. 

Since p is a complex number, it may be expressed in terms of the amplitude factor tan P, and the phase factor exp jA or, more commonly, in terms of just P and A. Thus measurements of P and A are related to the properties of matter via Fresnel coefficients derived from the boundary conditions of electromagnetic theory. ... 

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