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By Walsh J.L.

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Sample text

Indeed, if we were to replace the requirement that  be infinitely differentiable by the requirement that  be analytic, we would obtain only the zero function. Here we recall that a function  is said to be analytic on X if for every a 2 X ,  is given by a power series about a that is convergent on some neighborhood of a. This implies that  is of class C 1 and that the power series of  about a equals the Taylor series of  at a. 22 2 Test Functions Furthermore, an open set X in Rn is said to be connected if X is not the union of two disjoint nonempty open subsets of X (for more details, refer to [7, Sect.

The only problem is the differentiability at 0; see Fig. 2. From the power series for the exponential function one obtains, for every n 2 N, the estimate e y  yn for all y  0. 0/ D 0. 0/ D 0. 8. R/ be as in the preceding lemma. Let a and b 2 R with a < b. R/ with ˇ > 0 on  a; b Œ and supp ˇ D Œ a; b . x/ dx D 1. ˇ / ˇ has the same properties as ˇ (see Fig. 598 1 0 1 1 2 2 2 Fig. 9. R/ as in the preceding lemma, for 1  j  n. x1 ; : : : ; xn / 2 Rn . For a and b 2 Rn , define the function a;b W Rn !

0/ D 0. 8. R/ be as in the preceding lemma. Let a and b 2 R with a < b. R/ with ˇ > 0 on  a; b Œ and supp ˇ D Œ a; b . x/ dx D 1. ˇ / ˇ has the same properties as ˇ (see Fig. 598 1 0 1 1 2 2 2 Fig. 9. R/ as in the preceding lemma, for 1  j  n. x1 ; : : : ; xn / 2 Rn . For a and b 2 Rn , define the function a;b W Rn ! R by (see Fig. x/ dx  ; D 1: For a complex number c, the notation c  0 means that c is a nonnegative real number. x/  0 for every x 24 2 Test Functions Fig. 3 Graph of . 9 in the domain space of f .

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An Expansion of Meromorphic Functions by Walsh J.L.


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