Gamma-function

This is also a gamma function and may be solved with the help of a table of integrals. Evaluation of the integral gives the simple result... 

Combine the last two expressions and integrate to express in terms of M and a. The integrals are standard forms and are listed in integral tables as gamma functions. 

This integral is a gamma function and is readily solved using a table of integrals. In writing the last result uIq has been replaced by rQ the mean-square coil dimensions under 0 conditions. Equation (9.49) involves rjj" not r so we note that rQ = nlQ and replace n by I i - j I, the number of units separating units i and j, to obtain... 

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed tables. NOTE F(/i -1- 1) = Iq x e dx (gamma function) /(f) = Bessel function of the first land of order n. 

The mass distribution from the idealized MSMPR crystallizer is thus a Gamma function, as shown in Figure 3.8b. 

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)... 

After substitution of (A3.1) into (6.14), several integrals of the same type must be calculated. These integrals can be expressed via the degenerate hypergeometrical function d>(-, -, -) and gamma-function T( ) ... 

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. 

The gamma function is a generalization of the factorial introduced in Section 1.4. There, toe notation n = X- 2-3-4-was employed, with n a positive integer (or zero). The gamma function in this case is chosen so that r(n) = (n -1) . However, a general definition due to Euler states that... 

Several properties of toe gamma function follow from this definition, e.g. 

Other integral transforms are obtained with the use of the kernels e" or xk among the infinite number of possibilities. The former yields the Laplace transform, which is of particular importance in the analysis of electrical circuits and the solution of certain differential equations. The latter was already introduced in the discussion of the gamma function (Section 5.5.4). 

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