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The Gamma Function (Factorial Function) CHAPTER 8 THE GAMMA FUNCTION (FACTORIAL FUNCTION) The gamma function appears occasionally in physical problems such as the normalization of Coulomb wave functions and the computation of probabilities in statistical mechanics. In general, however, it has less direct physical application and interpretation than, say, the Legendre and Bessel functions of Chapters 11 and 12. Rather, its importance stems from its usefulness in developing other functions that have direct physical application. The gamma function, therefore, is included here. 8.1 DEFINITIONS,SIMPLE PROPERTIES At least three different, convenient definitions of the gamma function are in common use. Our first task is to state these definitions, to develop some simple, direct consequences, and to show the equivalence of the three forms. Infinite Limit (Euler) The first definition, named after Euler, is 1 2 3 n Ŵ(z) lim · · ··· nz,z0, 1, 2, 3,.... (8.1) ≡ n z(z 1)(z 2) (z n) = − − − →∞ + + ··· + This definition of Ŵ(z) is useful in developing the Weierstrass infinite-product form of Ŵ(z), Eq. (8.16), and in obtaining the derivative of ln Ŵ(z) (Section 8.2). Here and else- 499 500 Chapter 8 Gamma–Factorial Function where in this chapter z may be either real or complex. Replacing z with z 1, we have + 1 2 3 n z 1 Ŵ(z 1) lim · · ··· n + + = n (z 1)(z 2)(z 3) (z n 1) →∞ + + + ··· + + nz 1 2 3 n lim · · ··· nz = n z n 1 · z(z 1)(z 2) (z n) →∞ + + + + ··· + zŴ(z). (8.2) = This is the basic functional relation for the gamma function. It should be noted that it is a difference equation. It has been shown that the gamma function is one of a general class of functions that do not satisfy any differential equation with rational coefficients. Specifically, the gamma function is one of the very few functions of mathematical physics that does not satisfy either the hypergeometric differential equation (Section 13.4) or the confluent hypergeometric equation (Section 13.5). Also, from the definition, 1 2 3 n Ŵ(1) lim · · ··· n 1. (8.3) = n 1 2 3 n(n 1) = →∞ · · ··· + Now, application of Eq. (8.2) gives Ŵ(2) 1, = Ŵ(3) 2Ŵ(2) 2,... (8.4) = = Ŵ(n) 1 2 3 (n 1) (n 1) . = · · ··· − = − ! Definite Integral (Euler) A second definition, also frequently called the Euler integral, is ∞ t z 1 Ŵ(z) e− t − dt, (z) > 0. (8.5) ≡ ℜ 0 The restriction on z is necessary to avoid divergence of the integral. When the gamma function does appear in physical problems, it is often in this form or some variation, such as ∞ t2 2z 1 Ŵ(z) 2 e− t − dt, (z) > 0. (8.6) = ℜ 0 1 z 1 1 − Ŵ(z) ln dt, (z) > 0. (8.7) = t ℜ 0 When z 1 , Eq. (8.6) is just the Gauss error integral, and we have the interesting result = 2 Ŵ 1 √π. (8.8) 2 = Generalizations of Eq. (8.6), the Gaussian integrals, are considered in Exercise 8.1.11. This definite integral form of Ŵ(z), Eq. (8.5), leads to the beta function, Section 8.4. 8.1 Definitions, Simple Properties 501 To show the equivalence of these two definitions, Eqs. (8.1) and (8.5), consider the function of two variables n n t z 1 F(z,n) 1 t − dt, (z) > 0, (8.9) = − n ℜ 0 with n a positive integer.1 Since n t t lim 1 e− , (8.10) n − n ≡ →∞ from the definition of the exponential ∞ t z 1 lim F(z,n) F(z, ) e− t − dt Ŵ(z) (8.11) n = ∞ = ≡ →∞ 0 by Eq. (8.5). Returning to F(z,n), we evaluate it in successive integrations by parts. For convenience let u t/n. Then = 1 z n z 1 F(z,n) n (1 u) u − du. (8.12) = − 0 Integrating by parts, we obtain z 1 1 F(z,n) n u n n 1 z (1 u) (1 u) − u du. (8.13) nz = − z + z − 0 0 Repeating this with the integrated part vanishing at both endpoints each time, we finally get 1 z n(n 1) 1 z n 1 F(z,n) n − ··· u + − du = z(z 1) (z n 1) + ··· + − 0 1 2 3 n · · ··· nz. (8.14) = z(z 1)(z 2) (z n) + + ··· + This is identical with the expression on the right side of Eq. (8.1). Hence lim F(z,n) F(z, ) Ŵ(z), (8.15) n →∞ = ∞ ≡ by Eq. (8.1), completing the proof. Infinite Product (Weierstrass) The third definition (Weierstrass’ form) is 1 γz ∞ z z/n ze 1 e− , (8.16) Ŵ(z) ≡ + n n 1 #= 1The form of F(z,n)is suggested by the beta function (compare Eq. (8.60)). 502 Chapter 8 Gamma–Factorial Function where γ is the Euler–Mascheroni constant, γ 0.5772156619 .... (8.17) = This infinite-product form may be used to develop the reflection identity, Eq. (8.23), and applied in the exercises, such as Exercise 8.1.17. This form can be derived from the original definition (Eq. (8.1)) by rewriting it as n 1 1 2 3 n 1 z − Ŵ(z) lim · · ··· nz lim 1 nz. (8.18) = n z(z 1) (z n) = n z + m →∞ →∞ m 1 + ··· + #= Inverting Eq. (8.18) and using z ( ln n)z n− e − , (8.19) = we obtain n 1 ( ln n)z z z lim e − 1 . (8.20) Ŵ(z) = n + m →∞ m 1 #= Multiplying and dividing by 1 1 1 n exp 1 z ez/m, (8.21) + 2 + 3 +···+ n = m 1 #= we get 1 1 1 1 z lim exp 1 ln n z Ŵ(z) = n + 2 + 3 +···+ n − →∞ n z z/m lim 1 e− . (8.22) × n + m →∞ m 1 #= As shown in Section 5.2, the parenthesis in the exponent approaches a limit, namely γ ,the Euler–Mascheroni constant. Hence Eq. (8.16) follows. It was shown in Section 5.11 that the Weierstrass infinite-product definition of Ŵ(z) led directly to an important identity, π Ŵ(z)Ŵ(1 z) . (8.23) − = sin zπ Alternatively, we can start from the product of Euler integrals, ∞ z s ∞ z t Ŵ(z 1)Ŵ(1 z) s e− ds t− e− dt + − = 0 0 ∞ z dv ∞ u πz v e− udu , = (v 1)2 = sin πz 0 + 0 transforming from the variables s,t to u s t,v s/t, as suggested by combining the exponentials and the powers in the integrands.= + The Jacobian= is 11 s t (v 1)2 J , 1 s +2 + =− 2 = t = u t − t 8.1 Definitions, Simple Properties 503 u where (v 1)t u. The integral 0∞ e− udu 1, while that over v may be derived by + = πz = contour integration, giving sin πz. This identity may also be derived by contour integration (Example 7.1.6 and Exer- 1 cises 7.1.18 and 7.1.19) and the beta function, Section 8.4. Setting z 2 in Eq. (8.23), we obtain = Ŵ 1 √π (8.24a) 2 = (taking the positive square root), in agreement with Eq. (8.8). Similarly one can establish Legendre’s duplication formula, 1 2z Ŵ(1 z)Ŵ z 2− √πŴ(2z 1). (8.24b) + + 2 = + The Weierstrass definition shows immediately that Ŵ(z) has simple poles at z 1 = 0, 1, 2, 3,...and that Ŵ(z) − has no poles in the finite complex plane, which means that−Ŵ(z)− has− no zeros. This[ behavior] may also be seen in Eq. (8.23), in which we note that π/(sin πz) is never equal to zero. Actually the infinite-product definition of Ŵ(z) may be derived from the Weierstrass 1 factorization theorem with the specification that Ŵ(z) − have simple zeros at z 0, 1, 2, 3,.... The Euler–Mascheroni constant[ is fixed] by requiring Ŵ(1) 1. See= also− the− products− expansions of entire functions in Section 7.1. = In probability theory the gamma distribution (probability density) is given by 1 α 1 x/β x − e− ,x>0 f(x) βαŴ(α) (8.24c) = 0,x0. ≤ α 1 The constant β Ŵ(α) − is chosen so that the total (integrated) probability will be unity. [ ] 3 For x E, kinetic energy, α 2 , and β kT , Eq. (8.24c) yields the classical Maxwell– Boltzmann→ statistics. → → Factorial Notation So far this discussion has been presented in terms of the classical notation. As pointed out by Jeffreys and others, the 1ofthez 1 exponent in our second definition (Eq. (8.5)) is a continual nuisance. Accordingly,− Eq.− (8.5) is sometimes rewritten as ∞ t z e− t dt z , (z) > 1, (8.25) ≡ ! ℜ − 0 to define a factorial function z . Occasionally we may still encounter Gauss’ notation, (z), for the factorial function: ! & (z) z Ŵ(z 1). (8.26) = != + The Ŵ notation is due to Legendre.# The factorial function of Eq. (8.25) is related to the gamma function by Ŵ(z) (z 1) or Ŵ(z 1) z . (8.27) = − ! + = ! 504 Chapter 8 Gamma–Factorial Function FIGURE 8.1 The factorial function — extension to negative arguments. If z n, a positive integer (Eq. (8.4)) shows that = z n 1 2 3 n, (8.28) != != · · ··· the familiar factorial. However, it should be noted that since z is now defined by Eq. (8.25) (or equivalently by Eq. (8.27)) the factorial function is no longer! limited to positive integral values of the argument (Fig.
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