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Where (£ι)+ 1-λ and (£ι)ΐ 1-Λ also have poles. , xn) is not itself singular at these values of λ. It will be seen that there is, however, no contradiction when it is recalled that the Radon transform of a generalized function is defined only to within an unessential function. Although p^i)^1-* + />i(| 1 )l 1_A has poles at λ = k for nonnegative integers k, the leading term of its Laurent series about such a point is an unessential function, which we may therefore drop. , xn) for nonnegative integer k.

Thus the Radon transform of the generalized function defined by (5) is # ( £ , *o);1-A + />-(£. , (6) and />*(£, ^); 1 "* + P-{L x«)-1-* - ( - ^ 8 w [(f, * „ ) ] / In I * I for λ = * = 0, 1, ... , xj, both concentrated on the entire xx axis. , * Λ ) is l/> ΙΛI ^ |-X~A, for ΧΦ2Η (A = 0 , 1 , . . ) (7) (here | £x I-1-2*1 is an associated homogeneous generalized function). It may be noted that p2k\ ξχ ^1-^ is unessential if 2k < n — 1, and may therefore be dropped from Eq. (7') for these k. 4 The Radon Transform of Generalized Functions 31 (£r2fe sgn £x is an associated homogeneous generalized function).

Thus there still remains some arbitrariness in the generalized areas of the external triangles and angles. It will be seen later that this arbitrari­ ness associated with the a{ cannot be removed and that their variation introduces only unessential functions. These c^ are constants, or rather are independent of the intersecting plane. To see this, let φ^ξ, p) be the Radon transform of the charac­ teristic function of one of the octants. Now let us perform a dilation in one of the coordinate directions, say the x1 direction: χ ι = ^ι^ > 0)» x 2 = x 2 > x'z = xs · This changes φ 1 (^ 1 , ξ2 , £8 ; p) into λ-1φ1(λ-1ξ1, ξ2, ξζ ; p) [see Eq.

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