Incomplete gamma functions

As it is apparent from Eqs. (8) and (9), the decay of the errors with the truncation radii in the series (1) and (2) depends only marginally on the energy provided the conditions (7) are verified and it is determined essentially by the incomplete gamma Amctions. Thus, we can impose both the truncation errors to be as close as possible, simply by equating the arguments of both gamma functions. Thus, putting in Eqs. (8) and (9)... 

This integral is a special function related to the incomplete gamma function. The solution can be considered to be analytical even though the function may be unfamiliar. Figure 8.1 illustrates the behavior of Equation (8.8) as compared with CSTRs, PFRs, and laminar flow reactors with diffusion. 

The discretization error Cd for finite integration limits yi and y2 contains in addition to (D.8) two extra terms (under the sum) that contain incomplete Gamma functions. We don t need their explicit form for the estimation of the dominating part of the overall error. Of course, expanding these extra terms in powers of h would lead to the error estimation (A.4), that holds for extremely small h (and sufficiently small /) which is rather irrelevant in the present context. 

Note that in Eq. (9.11) the coefficient in the external field is factored out. A form very similar to Eq. (9.11) has been employed by Abell and Funabashi (1973) in which the expansion coefficients are expressed as products of incomplete gamma functions of order k,—that is,... 

Stokes and Nauman, 1970), where the integral is an incomplete gamma function (the upper limit is finite) that can be evaluated by the E-Z Solve software (file exl9-7.msp). It has the same variance as given in 19.4-37. Equation 19.4-38 can also be used to estimate t and N from tracer data obtained by a step input, using the nonlinear regression capabilities of E-Z Solve (see Example 19-9 for further discussion of this technique). 

Here a = X and y(a, z) = Jo dt e is the incomplete gamma function. It can be noted that for the axially symmetric potential with a longitudinal field, the only dependence on X is the trivial one in Xp, while in the nonaxially symmetric potential obtained with a transversal field the relaxation rate will strongly depend on X through F(oc), which is plotted in Figure 3.6. 

The gamma functions Ak(p) and Bj(pt) may be obtained by the use of recursion formulas an extensive tabulation is due to Flodmark (141). In the case of Slater orbitals of principal quantum number 4 or 6, application of Slater s rules leads to nonintegral powers of r in the radial wave function consequently, changing to spheroidal coordinates introduces A and B functions of nonintegral k values, that is, incomplete gamma functions. These functions can, however, be computed (56, 57) and the overlap... 

An analytic expression for the reduced creep function, obtained from Eq. (3.11) with Eq. (3.6) for the linear array, is shown in Table 4 Eq. (T4). Je is the elastic equilibrium compliance and y (1/2, w) is the incomplete gamma function of order 1/2, extensively tabulated and available in most computer subroutine libraries. The last term accounts for contribution of plastic flow, where present. Reduced creep functions (recoverable and including flow contributions) are plotted in Fig. 4 as a function of Z. The... 

Calculation of the viscoelastic functions proceeds as above where, for example, Eq. (T 7) is the reduced relaxation modulus for the cubic array. The incomplete gamma function of order 5/2 may be obtained in simpler form through a recurrence relation and ... 

Unfortunately, the explicit solutions to these systems usually involve fairly complicated functions such as Bessel, Haenkel, theta, and incomplete gamma functions in which the arguments depend on the rate constants. The result is that these solutions are of use only if the ratios of the rate constants are known. Otherwise an inordinate amount of arithmetical labor is involved. 

When fc is a nonnegative integer, H(a, b, c, z) is easily evaluated in terms of the incomplete gamma function,... 

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