507. The expansional values of functions investigated in this and the preceding chapter enable us to evaluate limits of the form %. Thus x – loge (1+x) Example 1. when x=0. 1 - 3.x -(1-2)3 x-(x - 3x + 3.2003 - ...) The given fraction 1 - 3x – (1 - 3x + 3x2 – x3) 3x2 – 3x3+... _*-}x+... } when x=0. - 3x2 + x3 -3 + x 6A - 6 sin A-sin Example 2. when A=0. 0(3 – 4 cos 6+cos 20) 68 – 6(– 183 +12705 – ...)– (0–503 +...)3 The given fraction 38 - 40(1 - 102+2404 - ...)+(1 - 202 + 304 - ...) 105 - 2006+..._05+...+... = 1% when A=0. 345 - 145 +... 105 + ... 5+... 9 20+ + + + + + The hyperbolic functions. 508. We have 02 A4 46 3 (e® +e-0)=1+ 12 14 16 03 A5 07 } (eo – e-C) = 0 + 3. 15 17 The above expansions of (c + e-) and 1 (e®-e-) are called the Hyperbolic cosine and the Hyperbolic sine respectively of 0; and are written shortly cosh 0 and sinh 0. Thus cos 0 and sin 0 are obtained from cosh 6 and sinh 0 respectively by changing the signs of the alternate coefficients of the several powers of 0. sin 0 sinh 8 Or cos 0 and are obtained from cosh 0 and A 0 respectively by changing A2 into – 02. Further sinh 0 : cosh 0 is called tanh 0: and so on. 509. The following table of comparison between the Trigonometrical (or Circular) and the Hyperbolic functions should be verified and studied. It is assumed that a has any positive or negative value. Hyperbolic Functions. cosh (- x) = + cosh x. sin (-2)=- sin . sinh (-x)=- sinh c. cos 0 = 1. cosh ( =l. sin 0 = 0. sinh 0 = 0. sinh 2 = 1. Limit (x = 0) 1. = C + 1. + oc. + cos a periodically decreases cosh x always increases nuand increases: its values re- merically as x increases numericurring at period 27. cally. sin a periodically decreases sinh x always increases algeand increases: its values re- braically as ac increases algecurring at period 27. braically. cosa x + sino a = 1. cosh? 2 – sinh? ? 1. cos (x + y) cosh (c + y) y y. sin (c + y) sinh (c + 9) y y. These formulæ for the hyperbolic functions may be proved from their exponential or from their expansional equivalents. sinh 2 Y y 510. If cos x and sin had been defined by their expansions in x, it could easily be shown that coso x + sino x = 1. For, if each of the expansions is multiplied by itself and the two results added, + the coefficient of ac2n 1 1 1 1 1 (1 – 1)2 = 0, 2n 1.121 - 1 2 3 2n - 3 2n by the binomial theorem for a positive integer, sin y + + Hence the only term remaining in cos” x + sino x is 1. = 1. This latter result is, of course, obvious from the exponential equivalents of cosh x and sinh x. Geometrical Illustrations. 511. The student of Conic Sections should observe the following points; where we still define cos, sin, cosh, sinh by their expansions. The equation to an ellipse, referred to its semi-axes a and b, is 24/a2+y/b2=1. Hence we may put x=a cos and y=bsin 0, where o has a variable value for the several points of the ellipse. For, as shown in the last article, cos-p+sinp=1 for all values of 0. The equation to a hyperbola, similarly referred, is xola" - /b2=1. Hence we may put x=a cosh and y=bsinh ¢, where o has a variable value for the several points of the hyperbola. For, as shown in the last article, cosh - sinh2 p=1. 512. The student of the Integral Calculus should further observe that the area of the sector measured from 0 to any value o is |(xdy - ydx). d cos o Now d sin = cos o do ; sin φαφ, d sinhø=cosh dø; d cosh p=sinh o do, as may be easily seen by differentiating the expansions. cos =fab do=fabo. Hence in the ellipse ] | (ady – yds)= $ab s(c032 $+sin? d) dø In the hyperbola, (wdy – yds) = fab sccosh? $ – sinh? ") do = tabsdo=fabo. = Hence, in either conic, o denotes the ratio of the sectorial area to the triangle whose base and altitude are respectively the semi-axes. Putting a=b, we have the circle or the rectangular hyperbola. In the case of the circle we have shown (Art. 54) that $ (the circular measure)=the ratio of the sector to the triangle whose base and altitude are each equal to the radius. And we now see that a similar result holds for any central conic. cos O=1 + + + = + + 513. To express tan 0 and cot as continued fractions in terms of 0. By the above expansions of cos 0 and sin 0, we have A2 A6 .... A4 A6 2.3 2.4.3.5 2.4.6.3.5.7 Here we have arranged the denominators with the even factors before the odd. Now let f (n)= A6 1 2 (2n+1) * 2.4(2n+1)(2n+3) 2.4.6 (2n+1)(2n+3)(2n+5) ) A6 - 2 (2n-1) * 2.4(2n-1)(2n+1) 2.4.6 (2n-1)(2n+1)(2n+3) ( ) .. f(n-1)-f(n) = 44 -6° f (n+1) 2(2n+3) 2.4(2n+3)(2n+5) (2n-1)(2n+1) (2n-1) f (n-1) A2 (2n + 1) f(n) f(n+1) sin 0 0 A2 A2 A2 1 &c. 3-5-7, f (2) tan 0 1 1 A2 A2 A2 and &c. 0 A cot 1 3 5 7-' + (2n-1)(2n +1)[1-2(2n+3) + -...] § 3. RESOLUTION INTO FACTORS. Endless Factorisation. 514. The product of an infinite number of factors may be used with the same limitations as the sum of an infinite number of terms. Thus J. T. 24 DEF. An endless product is said to be convergent if the product of its factors may be made as near as we please to some one finite quantity by increasing the number of the factors. It is convenient to understand the word “finite' here to exclude zero as well as infinity, so that if an endless product is convergent so also is its reciprocal. Every factor may be assumed to be positive, since a change in sign of any one factor will not affect the magnitude of the product. 515. The properties of an endless product may always be investigated by taking its logarithm, by means of which it is reduced to an endless sum. Thus the values 0 and 0, which we exclude for convergency of a product, correspond respectively to log 0 and log , i.e. to -co and +, which we exclude for convergency of a sum. 516. Corresponding to the distinction between positive and negative in terms, is the distinction between greater and less than unity in factors. Thus Since the nth factor of a product is equal to the ratio of the product of n factors to that of n 1 factors, a product cannot be convergent unless, when n is infinite, the limit of the nth term is unity. (Cf. Art. 443.) A few of the more important theorems may be given. 517. A product whose factors are alternately greater and less than unity will be finite for all values of n if the product of each factor into the next is also alternately greater and less than unity. For consider the product v.v.V3 ... Vn..., where v,> 1; v,<1; ...; and V,V, > 1; V.V; < This product = V1V.. VzV4. VgV6 · and is ..> 1. And it also = V1. V.V3 • V4U5 and is .. <vj. (Cf. Art. 446.) <l; . ... 518. A product, all of whose factors are greater than unity, is convergent if the logarithm of every factor to the preceding as base is less than some quantity less than unity. |