1 5.The Gamma Function (Factorial Function ) 5.1 Definition, Simple Properties At least three different, convenient definitions of the gamma function are in common use * Infinite limit. The first definition, named after Euler, is (5.1) This definition is useful in developing the infinite-product form of Γ(z). Here z may be either real or complex. Replacing z with z+1, we have (5.2) Appears in normalization of Coulomb wave functions and the computation of probabilities in statistical mechanics

2 * Definite integral (Euler) This is the basic functional relation for G (z). Also from the definition (5.3) (5.4) A second definition, also frequently called Euler’s form, is (5.5)

3 The restriction on z is necessary to avoid divergence of the integral. When G (z) does appear in physical problems, it is often in this form or some variation such as When z=1/2, Eq.(5.6) is just Gauss error function, and we have the interesting result (5.8) (5.6) (5.7) To show the equivalence of these two definitions, consider the function of two variables (5.9) with n is a positive integer. Since (5.11)

4 by Eq.(5.) Returning to F(z, n), we evaluate it in successive integration by parts. For convenience let u= t/n. Then Integrating by parts, we obtain (5.12) (5.13) Repeating this, we finally get (5.14)

5 This is identical with the expression on the right side of Eq.(5.1). Hence (5.15) by Eq.(5.1), completing the proof The third definition is (5.16) where  is the Euler-Mascheroni constant, (5.17) * Infinite product This form can be derived from the original definition by writing it a

6 Inverting and using we obtain Multiplying and dividing by we get (5.18) (5.19) (5.20) (5.21)

7 * Factorial notation Eq.(5.5) can be rewritten as to define a factorial function z! The factorial function of Eq.(5.5) is, of course, related to the gamma function by (5.22) (5.25)

8 If z =n, a positive integer, we have However, it should be noted carefully that z! is now defined by (5.25) the factorial function is no longer limited to positive integer values of the argument (Fig.5.1). The difference relation (Eq.(5.2)) becomes (5.27) (5.28) (5.29)

9 This shows immediately that 0!=1 (5.30) for n, a negative integer. * Double factorial notation In many problems, we encounter products of the odd (or even) positive integers. For convenience these are given special labels as double factorial: (5.33b) Clearly, these are related to the regular factorial by (5.33c)

Digamma Functions As may be noted from the three definitions in the previous section, it is inconvenient to deal with the derivatives of the gamma or factorial function directly. Instead, it is customary to take the natural logarithm of the factorial function(Eq.(5.1)), convert the product to a sum, and then differentiate, that is Differentiating with respect to z, we obtain (5.36) (5.37) (5.38)

11 which defines F(z), the digamma function. From the definition of the Euler constant, the above equation may be rewritten as 5.3 The Beta Function Using the integral definition(Eq.(5.25)), we write the product of two factorials as the product of two integrals. To facilitate a change in variables, we take the integrals over a finite range. (5.39) (5.57a) Replacing u with x 2 and v with y 2, we obtain (5.57b)

12 Transforming to polar coordinates gives us The definite integral, together with the factor 2, has been named the beta function Equivalently, in terms of the gamma function (5.58) (5.59a) (5.59b)

13 * Definite integrals, alternative forms The beta function is useful in the evaluation of a wide variety of definite integrals. The substitution t=(cosθ) 2 converts Eq.(5.59) to Replacing t by x 2, we obtain The substitution in Eq.(5.60a) yields still another useful form (5.60a) (5.60b) (5.61)

14 * Verification of π a / sin π a = a ! (- a )! relation If we take m= a, n= -a, 0< a < 1, then On the other hand, the last equality is obtained by using the previous result for the contour integral in Chapter 2). Therefore, we have proven the relation. (5.62)

Incomplete Gamma functions Generalizing the integral definition of the gamma function, we define the incomplete gamma functions by the variable limit integrals and Clearly, two functions are related, for The choice of employing or is purely a matter of convenience. If the parameter a is a positive integer, Eq. (5.68) may be integrated completely to yield (5.68) (5.69)

16 * Error integrals They can be written as incomplete gamma functions a =1/2. The relations are (5.70) (5.8a) (5.8b)