Differential-operator integrals

The components of the operator P are hermitian.2 In general, any differential operator Q has a hermitian adjoint Qf, defined by the integral relation... 

Thermodynamic derivations and applications are closely associated with changes in properties of systems. It should not be too surprising, then, that the mathematics of differential and integral calculus are essential tools in the study of this subject. The following topics summarize the important concepts and mathematical operations that we will use. 

The mathematical notion of an operator may be unfamiliar it is a rule for modifying a function. A comparison of the ideas of operator and function may be useful Whereas a function acts to take an argument, called the independent variable, as input, and produces a value, called the dependent variable an operator takes 2l function as input and produces a function as output. Multiphcation of a function by a constant, taking a square or square root, differentiation or integration, are examples of operators. Table 8.1 contains examples of functions and operators. 

The special branch of mathematics known as numerical analysis has assumed an added importance with the extensive use of digital computers. Since these calculators perform only the fundamental operations of arithmetic, it is necessary that all other mathematical operations be reduced to these terms. From a superficial viewpoint it might be concluded that such operations as differentiation and integration are inherently better suited to analog computers. This is not necessarily true, however, and depends upon the requirements of the particular problem at hand. 

If At 6 E is a function of a real or complex parameter t, we can define differentiation and integration with respect to t, usual rules of operations being applicable to them. Also regularity (analyticity) of At can be defined and Cauchy s, Taylor s and Laurent s theorems are extended to these regular functions. 

The d1/2/df1/2 and d-1/2/df1/2 operators are respectively the semidifferentiation and semi-integration operators [81]. These are analogues of the familiar differentiation and integration operators of the calculus. Since they are unfamiliar to many chemists, Table 6 has been included to illustrate some of their definitions and properties. The semi-... 

Just as logarithms and exponentials are inverse operations, integration is the inverse of differentiation. The integral can be shown to be the area under the curve in the same sense that the derivative is the slope of the tangent to the curve. The most common applications of integrals in chemistry and physics are normalization (for example, adjusting a probability distribution so that the sum of all the probabilities is 1) and calculation of the expectation values of observable quantities. 

Equation (139) was already discussed elsewhere [22,23,31] as a phenomenological representation of the dynamic equation for the CC law. Thus, Eq. (139) shows that since the fractional differentiation and integration operators have a convolution form, it can be regarded as a consequence of the memory effect. A comprehensive discussion of the memory function (137) properties is presented in Refs. 22 and 23. Accordingly, Eq. (139) holds for some cooperative domain and describes the relaxation of an ensemble of microscopic units. Each unit has its own microscopic memory function m (t), which describes the interaction between this unit and the surroundings (interaction with the statistical reservoir). The main idea of such an interaction was introduced in Refs. 22 and 23 and suggests that mg(f) JT 8(f,- — t) (see Fig. 50). 

This text assumes a solid background in algebra. All of the mathematical operations required are described in Appendix One or are illustrated in worked-out examples. A knowledge of calculus is not required for use of this text. Differential and integral notions are used only where absolutely necessary and are explained where they are used. 

Becanse V is the volnme of the filtrate collected at any time t, Vs is (dVldt)IS also, expressing as cV (where c is the mass of cake collected per unit volume of filtrate), substituting in Equation (7.60), and integrating under the assumption of constant pressure differential operation yields... 

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