Remark on algorithm 282 [S22] derivatives of ex/x, cos(x)/x, and sin (x)/x
We now prove separately the inequality for every term of the summation. Let us assume the thesis is wrong > ([(XLk+:-Xi,k) 2 + (yLk+:-yl.~)~] t "Jr [(X2.k+x-X2,k) 2 + (y2,k+l-Y2,k)2]t). which is contradictory. We can have equality to zero iff i.e. if all the corresponding sides of the two polygons are parallel. (e, d) We must prove that the minimal polygon has at least one vertex on the boundary of the corresponding domain. In fact, consider a minimal polygon: there will be at least a couple of adjaeent noncollinear sides. The corresponding vertex must lie on the boundary, otherwise a shorter perimeter polygon could be found (see Figure 7 (a), (b)). Furthermore, from simple geometrical considerations it can be deduced that the normal to the boundary must bisect the angle between the two adjacent sides (Figure 7 (b)). Conversely, if a constraint is not active, i.e. a vertex is not on the boundary, the adjacent sides in the minimal polygon must be collinear. (e) From the convexity properties of the domain D and of the function f it can be deduced that: (i) all local minima are global; (ii) any convex linear combination of minima is a minimum as well; (iii) Jensen's relation (3) applied to two minima obviously holds with equality. We have proved in (b) that in our ease Jensen's relation holds with equality if and only if the corresponding sides of the two polygons are parallel. Now assume to have found a (global) minimum: any vertex between two noncollineax sides will satisfy the bisection property proved in (d). If no straight line segments axe present on the boundary of domain C (e.g. if domain C is a circle), it is not possible to translate two adjacent noncollinear sides still satisfying the bisection property. Therefore, all minimal polygons have the vertices between two noncollinear sides in the same position. Thus they differ only in the position of vertices corresponding to nonactive constraints: the minimal reduced polygon is unique. * The work forms part of a research program supported by the Bundesministerium fiir wissenschaftliche Forsehung and the Fritz ter Meer-Stiftung. c o m m e n t If a Laplace transform P(s) is given in the form of a real procedure, L/nv produces an approximate value Fa of the inverse F(t) at T. Fa is evaluated according to Fa =-~-~=: N must be even. Since the V~ depend on …