It then follows easily by induction that the integral definition of (x/b) is equivalent to the series definition afforded by equation (4). Now it is obvious that, if and consequently decreases steadily as n increases. It follows from the lemma of §2 that (-) R≥0. Moreover, in a similar manner, 0 ≤ (-)"13 R+1, p=(−)"'* {R,,,— Cr+1,pis} where 001; and this is the required result. p The proof that the series definition of c,, is equivalent to the definition as a coefficient in a Maclaurin expansion follows as in § 2. 4. As applications of the theorem proved in §§ 2, 3, we may take the following examples of functions whose Maclaurin expansions possess the property that the difference between the function and the sum of the r+1 terms of the expansion It is obvious from the integral definition that, if p is a positive integer, (xb) is a polynomial in x of degree p-1 with positive coefficients. has the same sign as, and is numerically less than, the (r+2)th term.* Now it may be assumed without loss of real generality that 0 < a <; and then we have i {coth (x + ia) — coth (x — ia)} *In the case of all the examples, the functions are really functions of x2, so the theorem holds for all real values of x. †The function under consideration is an odd function of a with period π. and since а <π-а<π+а <2π−а < ..., the requisite conditions are fulfilled. (V.) The functions (tanh (x + ia) + tanh (x — ia)}, i {tanh (x + ia) — tanh (x − ia)} may be obtained from (III.) and (IV.) by a suitable change in a, and consequently need no further consideration. (VI.) The theorem of §3 can easily be applied to the differential coefficients of the functions considered in (I.)—(V.) 5. The theorem proved in §§ 2, 3 is of importance in connexion with asymptotic expansions in three ways: (I.) If (t) be a function of the nature described in § 2, 00 then f(x)= [ "$(t)edt is an analytic function of x when 0 R()>0, and its asymptotic expansion may be obtained by the method of Borel. In the special case when x is positive and consequently, since le is positive, the asymptotic expansion of (x) possesses the property that the remainder after +1 terms has the same sign as, and is numerically less than, the (+2)th term. This property of F(x) is, of course, of importance when F(x) has to be calculated for assigned numerical values of x. (II.) The remainder after r terms, in the Euler-Maclaurin sum formula, F (a) + F (a + 1) + ... + F (b) = [° F (1) di a {F(m-1) (b)—F (3m-1) (a)}+R, and it can be shewn that R, has the same sign and is numerically less than the first term omitted if ((x) has a constant sign in the range a≤x≤b. If F(x) is a meromorphic function of the type considered in §§ 2, 3, the methods of this paper afford in some cases an alternative means of investigating the nature of the remainder. (III). In applying summatory formulæ of Plana's typet we have to consider integrals of the form where ¥ (t, x) p (t, x) dt, s" X (t, x) p (t, x) dí, and it is frequently of importance to know the sign of the remainder in the Maclaurin expansions of (t, x), X (t, x), qua functions of t. Since ¥ (t, x) + iX (t, x) = coth (t − ix) − 1, it follows from (III.) and (IV.) of §4 that the functions (t, x) and X (t, x), qua functions of t, satisfy the conditions of §2. * Bromwich, Infinite Series, p. 325. + Lindelöf, Le Calcul des Résidus, p. 58. ASYMPTOTIC FORMULE OCCURRING IN By G. A. SCHOTT and G. N. WATSON. 1. MODERN theories of the structure of the atom usually assume that the atom consists of a ring of electrons rotating, in the normal configuration of the atom, in a symmetrical manner round a small positive charge. A dynamical system, consisting of a number of equal masses uniformly distributed in a ring round a central nucleus, was made the starting-point of Clerk Maxwell's investigations* on Saturn's rings. It is well known that the mathematical discussion of the steady motion of both of these systems involves the consideration of the series where n is the number of elements of which the ring is composed. An asymptotic formula for S, when n is large, has already been investigated by us; † the formula in question is m m 2m πS~ Σ TM §„~ 2n {log ̧ (2n) + y − log,π} + ï (−)"B_πTM (21m — 2) m=1 m.(2m)! nm- and this formula is well adapted for calculating S when n> 5. There are various series of a similar nature which occur in the discussion of the disturbed motion of a ring of elements rotating round a central nucleus; the series are, however, more complicated than S, inasmuch as they depend on two variables, both of which are positive integers; these integers are generally called n and k. *Collected Papers, I., pp. 286-366. † Schott, Electromagnetic Radiation, p. 216; Watson, Phil. Mag., xxxi. (1916), pp. 111-118. |