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By Goong Chen; Jianxin Zhou

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The space of all distributions will be denoted by D (Rn ), or D . The Dirac delta function δ is now well defined as a distribution because δ , φ ≡ φ (0) is a continuous linear functional on D(RN ). We may add distributions or multiply them by C∞ functions to form new distributions. However, the product of two distributions is not well defined in general. Every L1loc (RN ) function f defines a distribution via f,φ ≡ RN f (x)φ (x) dx, ∀ φ ∈ D. 2) for a continuous linear functional T even when T is not an L1loc function.

We shall naturally write the distribution x−(2m+1) instead of |x|−(2m+1) sgn x.

And residues δ (k−1) (x)/(k − 1)! at λ = k. 19), but without Re λ being restricted to (0, 1): λ −λ |x|−λ = x− + + x− , λ −λ |x|−λ sgn x = x− + − x− . , pseudofunctions). 29) above that at λ = k both x− + and x− have poles, with respective residues (−1)(k−1) (k−1) 1 δ (x) and δ (k−1) (x). (k − 1)! (k − 1)! Thus |x|−λ has poles only at λ = 1, 3, 5, . , with residues 2 δ (2m) (x) at λ = 2m + 1, m ∈ Z+ . (2m)! At λ = 2m, m ∈ Z+ , the distribution |x|−λ is well defined, and for these values of λ we shall naturally write x−2m instead of |x|−2m .

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