By Menger K.
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Extra info for Algebra of Analysis
In the classical theory, the only differentiate (and even the only continuous) function satisfying the postulates 1 and 2 is the function associating ex with each x. From the two postulates we have derived that D exp • exp* In Chapter I we saw in the algebra of the exponential functions that exp c ^ 0 for each c. Hence D exp c ^ 0 for each c. Thus our postulate 2 concerning the exponential function implies the existence of a function which, in an al* gebra with a base of constants, justifies our conclusion DJ * 1 In the preceding section.
From now on we shall reserve the sym^ bol tan for this function given by the postulates 1. tan(f+g)- tanf+tang 1-tan f «tan g 2. D tan 0 * 1. For this function we have Q D tan • 1 4- tan m Q rec cos . In the classical analysis, for each constant a the function tan (a-x) satisfies postulate 1. The function associating tan x with z is the only one which satisfies postulates 1 and 2. In a paper "e and ir In Elementary Calculus11 (to appear in the near future ) we describe how the postulates D tan 0*1 and D exp 0 * 1 in conjunction with the functional equations for the tangential and exponential functions yield an intuitive introduction of ir and e, as well as a simple development of the ^natural* tangential and exponential functions ex and tan x (x measured in radians).
The Three Methods of An ti derivation, If in formula III' we replace f by Sh we obtain + IH^ Shg~ S(hg'Dg). ' The formula lU * is the source of two methods for the computation of antiderlvatives. The first of these methods consists in applying formula III* read from the right to the left, that is, in the form 3(hg*Dg) ~ Shg. In words: If the Integrand of an antiderl vative which we wish to find, can be represented as the product of what results from a function h by substitution of a function g times the derivative of this function g, then we obtain the 43 antiderivative we are looking for, by substituting g Into the antlderlvatlve of h.