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By Menger K.
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Additional resources for Algebra of Analysis
Logarithmic and Exponential Derivation. The formulae at 'the end of the two preceding sections can also be written as follows: 36 Df • f«D(logabs f) and Df • rec exp f*D(exp f). Replacing f In the former formula by a particular function f la called logarithmic derivation (or differentiation) of f. Similarly, replacing f In the latter formula by a particular function f might be called exponential derivation of f • We -apply the former. method with benefit whenever logabs f IB simpler than f. As an example of logarithmic derivation, we treat the power functions* From c - po m exp(c-log) it follows that log c-po « c*log which is Indeed simpler than c-po.
Gr) by p%• . ,rV Into non- vacuous, disjoint subsets.
Formulae of the Algebra of Derivation in the Notation of Antiderivation, The formulae A. - D. of the preceding section enable us to translate each formula of the Algebra of Derivation into a formula about antiderivation* We start translating the postulates I - III of the Algebra of Derivations Sf +Sg~ SD(Sf + Sg) ~ S(DSf + DSg) ~ S(f + g). f «g ~ SD(f «g)~ S(f *Dg +g-Df ) - S(f «Dg) +S(g«Df). fg - SD(fg) - S(Dfg«Dg). 42 We thus obtain I» S(f + g) ~Sf + Sg 1 f »g ~ S(f »Dg) + S(g-Df ) 1 fg~ S(Dfg-Dg).