1.5. DISCREPANCIES 13 As a result, since φ(k) t = t−k−1φ(k)(t−1·), for any a / −N we have ˜ a (φt) := (−1)k (a + 1) · · · (a + k) Fa+k φ(k) t = ta ˜ a (φ), i.e., ˜ a is still a homogeneous distribution. However, for a = −k with k ∈ N we have ˜ a (φt) = 1 (k − 1)! ⎛ ⎝t−k k j=1 φ(k−1)(0) j − t−k−1 ∞ 0 log(x) φ(k)(t−1 x) dx⎠ ⎞ for = 1 (k − 1)! ⎛ ⎝t−k k j=1 φ(k−1)(0) j − t−k ∞ 0 log(tx) φ(k)(x) dx⎠ ⎞ = t−k ˜ a (φ) + φ(k−1)(0) (k − 1)! log t = t−k ˜ a (φ) + Resa=−k ˜ a (φ) log t , so that the extended distribution is no longer homogeneous. A discrepancy arises with the loss of homogeneity of the extended homogeneous distribution at negative integers. Consequently, ˜ a is homogeneous whenever z → ˜ a+z (φ) is holomorphic at zero. 1.5.2. The extended Gamma function: obstruction to the functional equation. The extended Gamma function ˜ Γ obeys the following property for Re(b) 0: (1.18) ˜ b + 1) = b ˜ b), but a discrepancy arises since property (1.18) breaks down at nonpositive integers. Exercise 1.25. For any complex value b, show that ˜ b + 1) = b ˜ b) + Resz=0˜ b + z). Consequently, ˜ obeys the functional equation ˜ b + 1) = b ˜ b) outside the poles, but at a pole −k in Z≤0 we have ˜ −k + 1) = −k ˜ −k) + (−1)k k! . In both cases investigated in sections 1.5.1 and 1.5.2, the presence of a residue is responsible for an obstruction to the expected property.
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