The Gamma Function

The gamma function is one of a class of functions that is most conveniently defined hy a definite integral. Consider first the following integral, which can be evaluated exactly, 

A very useful trick is to take the derivative of an integral with respect to one of its parameters (not the variable of integration). Suppose we know the definite integral 

This operation is valid for all reasonably well-behaved functions. (For the derivative of a function of two variables with regard to one of these variables, we have written the partial derivative d/da in place of dIda. Partial derivatives will be dealt with more systematically in Chapter 10.) Applying this operation to the integral (6.68), we find 

Setting a = 1, now that its job is done, we wind up a neat integral formula forn  

This is certainly not the most convenient way to evaluate n , but suppose we replace n by a noninteger v. In conventional notation, this defines the gamma function  

The Gamma function was apparently first defined by the Swiss mathematician Euler. In terms of real variable x, it took a product form  

The notation Gamma function was first used by Legendre in 1814. From the infinite product, the usual integral form can be derived 

It is clear from this that r(l) = 1, and moreover, integration by parts shows 

It is clear that all negative integers eventually contain F(0), since 

Occasionally, the range of integration in Eq. 4.14 is not infinite, and this defines the incomplete Gamma function 

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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... 

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). 

It can be seen that the gamma function approximation of the compound density function is not accurate. However, as seen in Figure 6.3, the gamma approximation of the distribution function is sufficiently good, especially in the interval between 90 and 100% probability which provides the inventory needed to be able to serve at least 90% of the demand. 

As an alternative, equation 19.4-38 may be solved using the E-Z Solve software to obtain the concentration profiles. The gamma function can be evaluated by numerical integration using the user-defined functions gamma, fjrateqn, and rkint provided within the software. 

P5.02.10. GAUSS AND GAMMA FUNCTIONS WITH Gamma functions are E(tr) - 4 trexp(-2tr)... 

Closely related to the gamma function are the exponential-integral ei(a) defined by the equation... 

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