CHAPTER VI. ON EXPANSION OF FUNCTIONS. Section 1.—On functions of one variable. 129-3 The Theorems of Chapter IV afford rigorous proofs and limits of application of Taylor's and Maclaurin's Theorems, of the general truth of which perhaps little more than a favourable presumption can be raised from what is said in Chap. III. On referring to Art. 18, equation (6), it will be seen, that if d.v is an infinitesimal increment of x, we have by the definition of a derived-function, r(x + dx) — v(x) = r'(x) dx; (1) but if e>x or (as we shall, to preserve an uniform notation, say) h, is finite, see Art. 18, (7), F {X + h) - F (X) = h F'(x) -} Ri, (2) writing Ri for nh. Ri is therefore a function of h, which must be neglected when h is infinitesimal, but has a value finite and determinable when h is finite. Let us assume this function of A to be capable of expression in certain powers of h; then the problem which arises out of (2) takes the following form: Given that ¥(x) is continuous and finite for all values of x between x and x -+ h, it is required to expand F {x + A) in a series of ascending powers of A. We may also thus arrive at the equation (2) above. If r'(x) does not vanish, and remains finite and continuous for all values of x between x and x + h, then by equation (21), Art.116, F(# + A)-f(*) = h*{w+&h); as the right-hand member differs from h f'(x) by reason of the 6 which enters into r'(x + 6h), we may assume that hv'(x + 0h) =hr'(x) + Ri; so that we have r{x + h) —r(x) = hr'(x) + Ri We proceed to determine Ri: from (2), R, = v(x + h) — r(x) — hv'(x), and continuing in the same manner, if all the derived-functions of r(x) are finite and continuous between the assigned limits up to the nth inclusively, we have A A"-1 T(x + A) — r(x) — v\x) T — ... — T"-1(x) 1.2.3. and therefore A A2 A3 F(x + A) = F(«) + T'(x) J + *"(*) Y2 + *"*(*) f23 + "■■ - 1.2.8A.."(n-l) + oSb,'"(a?+<,A)- (8) This expression then gives the equivalent of F(a' + A) in terms of a series of ascending powers of A, and the conditions under which it has been formed indicate the cases in which the expansion is possible. As applications of this series have already been given in Article 73, it is unnecessary to add others. 130.] In equation (8) the equality of the two members is perfect, and the development may be considered as completely effected, except so far as some indeterminateness arises from the nature of 6, to which quantity a specific value cannot be assigned. It was however before shewn that it must be some positive and proper fraction; and sometimes, if the series is convergent, when n is very great, the last terms and their sum become infinitesimal, and we must neglect and we may write F(x + A) = r(x) + F'(tf) J + F'v) + *"'(*) ^3 + ... (9) in which incomplete form the series was first given by Dr. Taylor, and now generally bears his name. 131] In equation (8) let x = 0; that is, let us consider the function for all values of the variable between 0 and A; and let us write x for A, remembering that x is the superior limit; whereby the conditions are, that none of the functions or derived-functions are infinite or discontinuous for any value of x between 0 and x; then F(*) = r(0) + F'(0)| + F''(0)^ + + *-W 1.2/.".(Ll) + T^hn ■> this is Maclaurin's Theorem, of which an imperfect proof was given in Art. 57; and it accordingly appears that it is only a particular case of Taylor's. Many examples of this series having been given in Chapter III, it is unnecessary to add others. Of the general series (10) however the following are particular instances: Let it = 1, then r(x) = F(0) + Xv'(0x). (11) e* 1 Thus ex = 1 + xe>Jr, .-. = e*x. x sin* sin x = x cos Ox, — cos Ox. x l0*<1 + *> = lW xn 132.] If the series (10) is such that vn(6x) becomes infinitesimal as n increases without limit, then F(tf) = F(0) + x /(0) + F"(0) + ^ F"'(0) + (12) that is, the limit of the sum of the second member of the equation is T(x). 133.] Again, taking the other form of Taylor's Series, viz. (84) in'Art. 74, we have r(x) = F(a) + {x-a) F» + + + ifc^S)'"-1^ + T^T)n^ (13) the superior and inferior values of x in this case being respectively x and a; so that it is for all values of x between these limits that the conditions are to be satisfied. As particular cases of the formula we have r(x) = F(a) + (x — a) r'{a + 9(x—a)}, (14) T(x) = v(a) + (x-a)v'(a) + { ~g r"{a + 0(x-a)}. (15) 134.] Hence it appears that if we stop at the nth term of |