Logarithmic derivatives

We will often encounter the logarithmic derivative in this book. It provides a very handy way to derive various parameters of interest from rate expressions or thermodynamic equations. The logarithmic derivative of a function / with respect to a variable x is given by... 

The apparent activation energy of the reaction follows by taking the logarithmic derivative as expressed by Eq. (48) ... 

Thus, by taking the logarithmic derivative of Eq. (Ill), we find that the order is... 

Taking the logarithmic derivative of (A.36) with respect to r, multiplying on both sides by <0 S T 0>, and using finally the Liouville equation (2), we have (see A.33) ... 

Aden (1951) was apparently the first to introduce the logarithmic derivative... 

If the incident light is normal to the cylinder axis the scattering coefficients have their simplest form. However, the coefficients (8.30) and (8.32) are not in the form most suitable for computations. If we introduce the logarithmic derivative... 

Dn mx) in the coefficients (4.88) is computed by the downward recurrence relation (4.89) beginning with Z)NMX. Provided that NMX is sufficiently greater than NSTOP and mx, logarithmic derivatives of order less than NSTOP are remarkably insensitive to the choice of Z)NMX this is a consequence of the stability of the downward recurrence scheme for pn. For vastly different choices of Z)NMX, and a range of arguments mx, computed values of DNMX 5 were independent of Z)NMX. Thus, NMX is taken to be Max(NSTOP, mjc ) -I- 15 in BHMIE, and recurrence is begun with Z)NMX = 0.0 + z 0.0. 

The convergence criterion in BHCOAT is the same as that in BHMIE series are terminated after y + 4y1/3 + 2 terms. Unlike BHMIE, however, all functions, including logarithmic derivatives, are computed by upward recurrence it seemed pointless to compute these derivatives by downward recurrence when they are not the major obstacle to writing a program valid for an arbitrary coated sphere. 

As in the previous programs, series for scattering matrix elements and efficiencies are truncated after NSTOP terms, where NSTOP = x + 4x1/3 + 2. Gn(mx) is computed by (C.l) beginning with CNMX, successive lower-order logarithmic derivatives CNMXGx are computed by downward recurrence. Provided that NMX is sufficiently greater than NSTOP and mx, Gp for... 

Computation of Bessel Functions The Bessel functions Jn and Yn pose more computational problems than the logarithmic derivative. In BHCYL these functions are computed by an algorithm credited to Miller (British Association, 1952, p. xvii), further details of which are given by Stegun and Abramo-witz (1957) and by Goldstein and Thaler (1959) we outline this scheme in the following paragraph. 

Tests of BHCYL Computed values of J (x) and Yn(x) for various n and x were compared with values tabulated by Olver (1964). Computed logarithmic derivatives were also compared with values calculated from tabulated Bessel functions. In all instances there was agreement to as many decimal places as were given in the tables. This gives us some confidence that BHCYL does what it was designed to do. 

LOGARITHMIC DERIVATIVE G(J) CALCULATED BY DOWNWARD RECURRENCE BEGINNING WITH INITIAL VALUE 0.0 + I 0 - 0 AT J = NMX... 

This can be expressed as the ratio of dimensionless logarithmic derivatives, namely... 

Table 7.1 Band parameters for the 4d transition metals at their equilibrium atomic volumes. W is the logarithmic derivative of the bandwidth with respect to volume. (From Pettifor (1977).)...

Noting that the energy levels Ea are independent of T, we obtain the desired logarithmic derivative... 

In this section, we arrive at the quantization condition expressed in terms of periodic orbits. The periodic-orbit contribution to the trace formula can be written as the logarithmic derivative of a so-called zeta function,... 

Because of the logarithmic derivative, the poles of the resolvent appear at the zeros of the zeta functions so that we obtain the quantization condition... 

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