By Stefan Hildebrandt

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**Extra resources for Analysis 2 (Springer-Lehrbuch)**

**Sample text**

O u,,,' is less than 2e if j and k are large enough so that, in particular, on denoting by p the larger of the two numbers j and k, the non-negative number s - o, is less than 2e if p is sufficiently large, proving that a = s. Turning now to the rows of our - X oo matrix we observe that any partial sum EM-4 um" of the infinite series furnished by the elements in the (n + 1)st row of this matrix is dominated by o, where p is the greater of the two integers n and j and this implies that E" um" is dominated by a = s no matter what are the integers n and j so that the infinite series -a um" is convergent, with sum ~~
~~

We proceed to investigate what properties of a given piecewise continuous right-sided function h(t) are sufficient to ensure that h(t) is of exponential type. /p"+') + ... mite series converges absolutely at p = R + S where S is 40 41 The Laplace Transformation any positive number, and it may also converge (not, necessarily, absolutely) at p = R. 4, a... _ (-1)'" f I r/f coo' 8d9 and 2 I .. 3 2p1 . 4) .. , being alternating, is convergent. 4p') - .. which converges, with the sum (1 + pt)-*, over the neighborhood I p I > 1 of p = eo, also converges at p = 1.

Sided functions h*(t) - hi (t), h*(t) + h2'(t), h*(t) - h,'(t) and, since h'(t) is the linear combination 4 {h*(t) + h1'(t)I - 4 {h*(t) - h1'(t)l + {h*(t) + h2'(t)) -2{h*(t)-h2'(t)} of the four non-negative real-valued right-sided functions h*(t) t hi (t), h*(t) f h2'(t), it follows that the Laplace Transform, over the half-plane c > R, of h'(t) is the sum of the infinite power series (ao/p) + (a,/p2) + . In other words, the two right-sided functions h(t) and h'(t) possess, over the half-plane c > R, coincident Laplace Transforms, and this implies that h(t) coincides with h' (t) at any point t which is a continuity point of both h(t) and h'(t).