Square root of a complex

It follows that one square root of a complex number (where = ) is given by ... 

The square root of a complex number is a number that will yield the first number when multiplied by itself. Just as with real numbers, there are two square roots of a complex number. If z = re, one of the square roots is given by... 

When the dielectric function itself is complex, we must take the square root of a complex number or n = Ve where the tildes represent complex quantities. We can write... 

Neither of equations (iii) or (iv) are solutions to equation (7.46). However, if n was such that n2 was negative, then both functions would be solutions to the equation. This would require us to define the square root of a negative number, which is at odds with our understanding of what constitutes a real number. In Chapter 2, Volume 2, we extend the concept of the number to include so-called imaginary and complex numbers, which embrace the idea that the square root of a negative number can be defined. 

Complex numbers owe their origin to the quest for the square root of a negative number. Thus the so-called imaginary number i = is a fundamental element of complex numbers, written as z = X + iy, in which x is the real part and y is the imaginary part. Although real numbers quantify physical quantities, complex numbers provide very convenient representations of many physical phenomena. In quantum mechanics, the wave function is a complex function. Two-dimensional, incompressible, irrotational flows are represented by a complex flow potential, w = 9 h- t /, with 9, the velocity potential, as the real part, and /, the stream function, as the imaginary part. 

First step Find a particular solution u = v = v = vq + i v + j V2 which has also a vanishing fourth component. Since vv = v2 we may use equation (11), which was developed for the complex square root, also for the square root of a quaternion ... 

Surprisingly this complex sequence of events results in a first order rate expression for the overall conversion. However, the experimental rate constant of this rate expression consists of a square root of a quotient of products of elementary rate constants. This fact notwithstanding, the experimental rate constant is usually subject to limits in A and E, similar to those described above for elementary rate constants. For example, an analysis of the experimental frequency factor shows why it should be in the expected range of magnitudes. The experimental frequency factor is ... 

The singular values of a complex n x m matrix A, denoted by cr,(A) are the nonnegative square-roots of the eigenvalues of A A ordered such that... 

The determination of the number of the SHG active complex cations from the corresponding SHG intensity and thus the surface charge density, a°, is not possible because the values of the molecular second-order nonlinear electrical polarizability, a , and molecular orientation, T), of the SHG active complex cation and its distribution at the membrane surface are not known [see Eq. (3)]. Although the formation of an SHG active monolayer seems not to be the only possible explanation, we used the following method to estimate the surface charge density from the SHG results since the square root of the SHG intensity, is proportional to the number of SHG active cation com-... 

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... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

A unitary matrix may therefore be considered a kind of square root of unity, often complex-valued. Unitary matrices with all real elements are called orthogonal O, and satisfy a property analogous to (S9.1-12) ... 

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