2 1. GAMMA FUNCTION EXTENDED TO NONPOSITIVE INTEGER POINTS For a complex number a with real part Re(a) larger than −1, we can therefore consider the distribution Fa : S(R+) −→ C φ −→ 0 xa φ(x) dx. Fa defines a homogeneous distribution of degree a in the following sense. Given a Schwartz function f in S(R+) and any positive number t, we set φt = t−1φ(t−1·). Then, (1.1) Fa(φt) = ta Fa(φ) if Re(a) −1. The integral Γ(b) := 0 xb−1 e−x dx, which is defined for Re(b) 0, is called the Gamma function. The Gamma function z Γ(z) defines a holomorphic map on the half-plane Re(z) 0. Exercise 1.4. Check this assertion. Integration by parts shows that Γ(b) := Γ(b+1) b for Re(b) 0. Exercise 1.5. For any complex number b with positive real part show that: (1.2) Γ(b) := 1 b(b + 1) · · · (b + k 1) Γ(b + k) ∀k N. Deduce that Γ(k) = (k 1)! ∀k N. Extending the Gamma function to the whole complex plane is related to the problem of extending homogeneous distributions Fa considered by Hadamard and Riesz (see e.g. [Sch, Chapter II]) to all complex values a. One wants to assign to the poles of Γ a finite value, which amounts to assigning a finite value ˜ −k (φ) to negative integers −k. Exercise 1.6. Show that the map x (log x) e−x lies in L1(R+). Hence we can define the Euler’s constant1 γ := 0 log x e−x dx. The following elementary properties of the Gamma function are useful for forth- coming applications. Proposition 1.7. (1) The Gamma function is differentiable at any pos- itive integer k and (1.3) Γ (1) = −γ Γ (k) Γ(k) = k−1 j=1 1 j γ⎠ ∀k N {1}. 1 The Gamma constant was first introduced by Euler in 1735 as the limit γ = lim N→∞ N n=1 1 n N 1 1 t dt .
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