Page:EB1911 - Volume 27.djvu/296

Rh 27. We shall now obtain expressions for sin x and cos x as infinite products of rational factors. We have Factorlza- sin x =2 sin;~csin3%'=2'* siniisinii lion of sm 4 4 andCoslne. ~ x'f“21f - x'l'3"f S111 T-SIHTQ proceeding continually in this way with each factor, we obtain x. x+1r. x+21r x+n-r1r in x==2"'1Sll'l 'Sl[1°*°'l -. .smi- n rr S n n n where n is any positive integral power of 2. Now x-l-r1r x+n-r1r  x-I-r-/r. nr-x rvr. x in sm =sxn sm = in”—sinh S n n n n S n rt x-l-émr x nd sin-i=cosa n n

Hence the above may be written x. 1r. x 2-rr. x sm x=2"" sm; sm' 5-smz 5)  (L siwn) (I sin* x/n sin* m+11r/n sin* m+21r/n " ' SUI2 k"/11 and m is any fixed integer independent of n. It is necessary, when we make n infinite, to determine the limiting value of the quantity sin x . sin xy R, then, since the limit of n Sin x/n COS x/n is? and that of Sin mt/n is unit we have m1r/n y sin x  <   (lim R. -T - 1-T2 I-2,72   I-WT, mfw The modulus of R-1 is less than 1+-1.=e;=—>    -1 sin' m+I1r/n sinz m+2n/n ' ' ' S1112 IQ#/rl where p=mod. sin x/n. Now eAP2>I+Ap“, if A is positive; hence mod. (R-I) is less than exp. p2(cosec' rn-f-Ivr/11+  -lcosec kr/n)-I, or than exp. § p2n'{r/ m+1)2+. .. +I/k“}-I, or than exp. {p'n'/4m'}-I. Now p2=sin2 a./n.cosh2 B/n+cos2 a./n. sinh' B/n, if x=a+1;6; or p'=sin2 u./rt-l-sinh” /3/n. Hence lim, ,=, p'n'=a'-H82, hmnm, pn=mod. x. It follows that mod, ,=° (R-I) is between 0 and exp. {(mod 4)2/ xm”}-I, and the latter may be made arbitrarily small by taking m large enough. It has now been shown that sin x=x(r-x2/1r')(I-x2/2”1r2) (1 -:cz/m11r') (I -l-em), where mod. em decreases indefinitely as rn is increased indefinitely. When m is indefinitely increased this becomes

sin x=x<1-%) <1-é ...=x;l (1-7% ., (25)

This has been shown to hold for any real or complex value of x. The expression for cos x in factors may be found in a similar manner by means of the equation cos x=2 sin #cos W, or may be deduced thus

P (ae) C0Sx=srr 2x= nit = I 4x“> (I 4x”> (I i1, x2> 2 sm x P (x :i) rr” 3272 5212 nh- ggas xg

=5=0 I-Z-J-2n+I), "2 (26)

If we change x into tx, we have the formulae for sinh x, cosh x as infinite products—

sinhx x=0 I+-174; coshx 5:0 1+(2n+I),1r,

In the formula for sin x as an infinite product put x=, we after 2n factors in the numerator and denominator, we obtain the approximate equation

then get I  if we stop 2:e z| 2 Hz =§ ~<2"+'> or  =Vnw, where n is a large integer. This expression was obtained in a quite different manner by Wallis (Arithmetica infinitorum, vol. i. of Opp.)

28. We have

I @.' Series for sin (x-ty) = (x+y)P (I + mr cor, cone, sinx xp  ' gxrand or cos y-l-sin y cot x 3' 3/ 3' 3' 5 = (HT) (HW-l-rr)  (HW-21f

Equating the coefficients of the first power of y on both sides we obtain the series

cot x=$1⁄x$ +¢+¢+, $, +, i-, +. . (27)

From this we may deduce a corresponding series for cosec x, for, since cosec x=cot éx-cot x, we obtain

cosec x=$1⁄x$ − 12-36-l-1r x-1r+x+21r+x-21r x+31r x—31r+'" (28)

By resolving S(-éégi into factors we should obtain in a similar manner the series 2 2 2 2 2 2

tan x:1r-2x 1r+2x+31r-2x 3-/r+2x+51r -2x'51r-l-2x +"" (29)

and thence

secx-tan<;-l-5) -tan x-T 2x+T+2x-3T 2x-Q-556+ .... (30)

These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating. Glaisher has proved them by resolving the expressions for cos x/sin x and I/sin x  as products into partial fractions (see Quart. Journ. Math., vol. xvii.). The series for cot x may also be obtained by a continued use of the equation cot x=${cot %x+ cot %(x -li-1r)} (see a paper by Dr Schroter in Schl6milch's Zeitschrift, vo . xm.

Various series for may be derived from the series (27), (28), (29), (30), and from the series obtained by differentiating them one or more times. For example, in the formulae (27) and (28), by putting x=1r/rt we get I I I I Serlesior = lr  i  .. i. fr derived T 'mann I n-1+n+1 211-I+2n+1"' ' fmmgenyeg 1f 1 1 1 1 iorCotand ~"'" Smn I+n-1 n+I 2n-1+2n+1"' ' C°"'° If we put n=3, these become - I I 1 I I 1r-3x/3 i-E+;-E+;-§ +... sa/2;;; 1 1 1r- 2 I-l-2 4 54-7-l-S...

By differentiating (27) we get

cosec2 x=E+2+(x-1f>2+2+2+- - -1 putx=;:, and we get =9{ 1-i-$2-I-%+$+}

These series, among others, were given by Glaisher (Quart. fourrr. Math. vol. xii.).

29. We have sinh 1rx=1rxP <1 +;$>, cosh 7l'x=P' I+; if we differentiate these formulae after taking logarithms we obtain the series

/2x I I I 1 gigs, "

zx wth, rx 2x2 1'+x2+2'-I-x2+32-|-x2+' ' " ll' I I I 22 tanh  ...

These series were given by Kummer (in Crelle’s Journ. vol. xvii.) The sum of the more general series + . . ., has been found by Glaisher (Proc. Land. Math. Soc., vol. vii.)

If Um denotes the sum of the series $1⁄1^{m}$, +$1⁄2^{m}$+$1⁄3^{m}$+ ..., Vm that