New PDF release: Cours de mathématiques Tome 4 Equations différentielles

By J.Lelong-Ferrand, J.M.Arnaudiès

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Extra info for Cours de mathématiques Tome 4 Equations différentielles Intégrales multiples

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We now calculate the generalized areas of the exterior triangles and of the angles, not for one octant, but simultaneously for all the octants intersected by the plane. If the plane neither passes through the origin nor is parallel to one of the coordinate axes, it intersects seven octants in one triangle, three external triangles, and three angles. Let s be the area of the triangle, si be the generalized areas of the angles, and ai be the generalized areas of the external triangles (/ = 1, 2, 3; see Fig.

We shall make use of the following result, which we present here without proof. *-*. where #(λ), έ(λ), and c(A) are analytic functions of λ still to be determined. >-*. (5) We must now calculate a(X)> 6(λ), and c(X). Note first that the term whose coefficient is b(X) can not occur, so that b(X) == 0. This is because if p and | x have opposite signs, the (ξ, x) — p plane does not intersect the ξ\— ξ\— ξ\> 0 cone. Now to calculate a(X)9 consider a plane which intersects the cone in an ellipse, say the x1 = 1 plane; in other words we set ξ — ξ0 = (1, 0, 0) and p = 1.

It is, of course, possible to derive Eq. (1) by the limiting process described in the previous section. We shall use a different approach here, however, in which we first write down the Radon transform of the generalized function %ι)(4--4-4)λ for complex λ, and then obtain the limit as λ —► 0. For simplicity we shall treat the circular cone x\ — x\ — x\> 0, xx > 0; the result can then be generalized by affine transformation. We proceed from the following fact. )/*(*) = θ(Χι)(χΙ - X\ - Xl)\ is -h I P \2+2λΟ-λ~Η()< (2) Κ) ^ 2 ( λ + 1)8ίηττλ 1ΡΙ ^ - Κζ)' where ξ)(ξ) = ξ\ — ξ\ — ξ\ .

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