distribution facts
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A short collection of distribution-theory facts dated 6.18.09 and signed PhL, in the Stakgold folder. It lists delta sequences (Lorentzian, sin(kx)/(πx), sin^2(Rx)/(πRx^2)), a test function, the Fourier representation of delta, the Poisson-type sum of deltas, principal-part 1/x limits, and rules for translation, scaling and derivatives. Each item cites page numbers in volumes I and II, apparently of Stakgold.
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Collection of Distributional Facts PhL 6.18.09
1. Delta sequence: sk(x) = (1/π) k/(1 + k2x2) limk→∞ sk(x) = δ(x) // I p 19A
2. Delta sequence: sk(x) =sin(kx)/(πx) limk→∞ sk(x) = δ(x) // I p 27 A
3. Test function in S1: φ(x) = exp(1/[x2-1]) |x| < 1 // I p 29 A
4. Fact: δ(x) = (1/2π) ∫dy eiyx // I p 43 B
5. gn(x) = - cos(nx)/n limn→∞ gn(x) = 0 // I p 44 B
6. limn→∞ sin(nx) = 0 // I p 44 B
7. Fact: Σint k δ(x-k) = 1 + 2 Σn=1∞ cos(2nπx) // I p 45 1.22 and similar facts
8. d/dx log(|x|) = pf(1/x) < pf(1/x), φ> = (1/x) φ(x) // I p 49
9. limR→∞ (1-cos(Rx))/x = pf(1/x) // I p 49 E
10. limα→±0 1/(x+iα) = ∓ iπδ(x) + pf(1/x) // I p 50 1.27
11. Translation rule: < t(x-a), φ(x)> = < t(x), φ(x+a)> // I p 35
12. Scaling rule: < t(ax), φ(x)> = (1/|a|) < t(x), φ(x/a)> // I p 36
13. Derivative rule: <t(n)(x) ,φ(x)> = (-1)n <t(x) ,φ(n)(x)> // I p 37 1.19a
14. Fact: limR→∞ (1/π) sin2(Rx)/(Rx2) = δ(x) // II p 16 5.8