Lectures On the Theory of Functions of Real Variables 1
The book Lectures On the Theory of Functions of Real Variables 1 was written by author James Pierpont Here you can read free online of Lectures On the Theory of Functions of Real Variables 1 book, rate and share your impressions in comments. If you don't know what to write, just answer the question: Why is Lectures On the Theory of Functions of Real Variables 1 a good or bad book?
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w — 1 ! nl where A is independent of u. This function is obviously a con- tinuous function of u in 21 for any x in St. Differentiating with respect to u, we get, observing terms cancel in pairs, 9'(u:, = (^^ZJDTI s -/")(^) + AI (3 71—11 for any u tvithin 21. Thus the derivative of g{u') is finite within 21. To apply Rolle's theorem to g(u)-, with reference to the interval ^ = (a, a;), it is only necessary to determine A in 2) so that But obviously g(x')=0. We therefore suppose A so chosen that ...=f(x-) -fCa-) -(x- a)f\a^ - •.• - -fcl^)^/(-i)(«) - (^z:^ A. (4 w— 1 ! w 1 Then by Rolle's theorem, there is a point a(c)-^l = 0. (5 M— 1 1 As c^^a;, the first factor in 5) is not 0. Hence the parenthesis is 0, which gives r^\c)=A. Putting this value of A in 4), we have 1). TAYLOR'S DEVELOPMENT IN FINITE FORM 257 2. The formula 1) is called Taylor s development of f(cc) in finite form. It may also be written as follows : Set x = a+h, c = a-\-6h.
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