Integration variables

This is a very sharply peaked fiinction around f p with a width of the order of k T. (At T= is a step fiinction and its temperaPire derivative is a delta fiinction at p.) Thus in the integral for one can replace by its value at f p, transfomi the integration variable from to v and replace the lower limit of v, which is (-pEp), by (- ). Then one obtains... 

We note here that the qiiantnm levels denoted by the capital indices I and F may contain numerous energy eigenstates, i.e. are highly degenerate, and refer to chapter A3.4 for a more detailed discussion of these equations. The integration variable in equation (A3.13.9) is a = 7 j / Ic T. 

Variable Flow Rate Conventional variable clearance volume and valve lifting devices are impracticable at high pressures and, should it be necessary to vary the flow rate, use has to be made of variable speed electric drives or magnetic clutches. Integral steam and gas engines have been used and Burckhardt (168) developed an hydrauhc drive to provide an integrated variable capacity machine, but its efficiency is less than that of a straight mechanical drive. 

Here we have changed the integration variable x to s, which is defined by... 

Since the Vxc functional depends on the integration variables implicitly via the electron density, these integrals cannot be evaluated analytically, but must be generated by a numerical integration. 

As will be shown later (Section 1.11), this may be rewritten by an interchange of the integration variables ... 

We may, therefore, introduce a new integration variable kg = ik4, thus transforming the integral (11-676) into an integral over a euclidian four space... 

For further progress, it is convenient to change integration variable from x to F, eq. (3.1),... 

The variable p (r) denotes the nuclear charge density at a point r with coordinates r = xi,X2,x ), and V r) is the Coulomb potential set up at that point by all other charges (the Coulomb constant k = l/(47t o) is dropped in this description). The integration variable in (4.1) is the volume element dr = Ax dr2dx3. The origin of the coordinate system is chosen to coincide with the center of the nuclear charge. A more convenient expression can be obtained by expanding V r) at f = (0,0,0) in a Taylor series, that is,... 

The integral can be solved by conversion from Cartesian to spherical coordinates. Then, the integration variable takes the convenient form dr = r drsinddddcj), which yields... 

This coordinate change is analogous to changing integration variables x, y, z to spherical polar coordinates r, 0, 4>-... 

The inner product can be calculated using either / or g, or — f(x) or g(k) depending on the integration variable x or k. More general integrals can also be represented in this notation, e.g. 

In the second step, after substituting the expression (127) for y(x x ), four terms were obtained in the integrand, reduced subsequently to two terms only by renaming integration variables x - x, in two of four terms. Next, the Coulomb potential is expanded formally in a power series of r jr ... 

Note that the integral above is nominally a function of p + — 2 coordinates. Furthermore, because the functions of interest are antisymmetric, it does not matter which coordinates are chosen for the dummy integration variable z. 

A trace over coordinate x is indicated by connecting the line labeled n to the line labeled n. The labels n and n are then deleted, since these coordinates become a single dummy integration variable. Diagrammatically, this creates a loop in the case that both x and x are arguments of the same cumulant. As an example, we apply tr3 to Fq. (43) to obtain... 

X = time constant of the exponential modifier t = centre of gravity (top) of the Gaussian t = dummy integration variable... 

This sum rule for extinction is written more compactly if we transform the integration variable from frequency to wavelength and assume that the static dielectric function is real and finite ... 

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