4 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS 1.2. A meromorphic extension of the Gamma function Let a be a complex number, the real part of which satisfies Re(a) −1, and let φ be a Schwartz function on R+. Exercise 1.11. Show that Fa(φ) = Fa+1(φ ) a+1 . After integrating by parts k times we get (1.6) Fa(φ) = (−1)k (a + 1) · · · (a + k) Fa+k(φ(k)). Since φ(k) is a Schwartz function, the expression on the right-hand side is well defined provided a + k has its real part larger than −1. The expression (−1)k (a+1)···(a+k) Fa+k(φ(k)) therefore defines an extension of Fa to the half-plane Re(a) −k 1. Given a complex number a, there is a positive integer k such that Re(a) −k 1, and we set (1.7) ˜ a (φ) := (−1)k (a + 1) · · · (a + k) Fa+k(φ(k)). Exercise 1.12. Show that this definition does not depend on the choice of k −Re(a) 1 in checking that ˜ a (φ) = (−1)k+l (a+1)···(a+k+l) ˜ a+k+l (φ(k+l)) for any positive integer l. Equation (1.7) therefore extends Fa(φ) to a meromorphic function a ˜ a (φ) on the plane with simple poles at negative integers. The residue at a negative integer −k −N is given by Resa=−k ˜ a (φ) = lim a→−k (a + k) ˜ a (φ) = (−1)k (−k + 1) · · · (−1) 0 φ(k)(x) dx = φ(k−1)(0) (k 1)! , (1.8) where for a meromorphic function f with a simple pole at z0, we have set Resz=z 0 f := lim z→z0 ((z z0)f(z)) . When applied to φ : x e−x, this construction provides an extension of the Gamma function to the whole complex plane, defined on the half-plane Re(b) −k with k N by (1.9) ˜ b) := 1 b(b + 1) · · · (b + k 1) Γ(b + k). Exercise 1.13. Show that ˜ has simple poles at integers −k −N {0} with residue at these poles given by (1.10) Resb=−k˜ b) = (−1)k k! . From now on we use the same notation Γ for the extension ˜ By (1.9) we have the following recursive formula: (1.11) Γ(b + k) = b(b + 1) · · · (b + k 1) Γ(b) if Re(b) 0.
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