By J. Glimm, L. Gross, Harish-Chandra, R. V. Kadison, D. Ruelle, I. E. Segal, C. T. Taam

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The classical procedure for nding the nite part of divergent integrals appears ad hoc and leads to questions about the validity of the procedure. First, could the appearance of divergent integrals in applications be the result of errors in modeling the physics of the problem? Second, will the method lead to a unique analytical expression or do di erent analytical expressions lead to equivalent numerical results? The generalized function theory clearly answers these questions. Let us rst examine the function f (x) = lnjxj , which is locally integrable.

Ffowcs Williams, J. ; and Hawkings, D. : Sound Generation by Turbulence and Surfaces in Arbitrary Motion. Philos. Trans. R. Soc. London, ser. A, vol. 264, no. 1151, May 8, 1969, pp. 321{342. 42. : Can Shock Waves on Helicopter Rotors Generate Noise? A Study of the Quadrupole Source. , vol. 1, 1990, pp. 323{346. 43. ; and Lee, Yung-Jang: Development of a Shock Noise Prediction Code for High-Speed Helicopters|The Subsonically Moving Shock. , vol. 2, 1991, pp. 773{790. 44. ; and Myers, M. : An Analysis of the Quadrupole Noise Source of High Speed Rotating Blades.

The momentum equation gives @ @t (ui ) + @ 0  ui uj @xj 1 + Pij = Pij nj  (f ) (4:60) where Pij = E ij + (p 0 p0)ij is the compressive stress tensor and Eij is the viscous stress tensor. 60), subtract the latter from the former, and nally subtract c 2 @ =@ xi2 from both sides to get 2 0 p = @ 2 @ xi @ xj 2 3 Ti j h(f ) 2 0 @@x Pij nj  (f ) i 3 + @ @t [0 vn (f )] (4:61) where p0 = c2 ( 0 0). Here Ti j is the Lighthill stress tensor. Now we have added h(f ), the Heaviside function, on the right side to indicate that Tij 6= 0 outside the surface f = 0.

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