E(z)=limm=∞ρm(cos m¢ +i sin m4>)l, by De Moivre's theorem. SiI'lC€ p"'= (I m§ I+ im, W8 have llITlm...mp'” =e'. 1im, ,. .¢, § i-4-m '/mix/vm Jim. Let r be a fixed number X . y? im
ess than x/m-l-x//rn, then lim, ,, ,, , 2 I+m('/m+x'/m 2 hes 2
between I and lim, ,., , I 4% g my or between I and 51/2 /212; hence since r can be taken arbitrarily large, the limit is I. The limit of m¢ or m tan'1{y/'(x+m)} is the same as that of my/(x+m) which is y. Hence we have shown that E(z) =e'(cos y-I-i sin y).
21. Since E(x-I-iy)=1§ '(cos y-i-sin y, we have cos y-I-i sin y =E(iy), and cos y-i sin y=E(-iy). Therefore cos y=%{E(iy) -l-E(-iy)}, sin y=§ i{E(iy)-E(-iy)l¥ and using ExP°"°"“"I the series defined by E('iy) and E(-fy), we find that V"'f""s °f cos y = I - y”/2! + y*/4! - ..., sin y = y -y“/3! T"'g";;'°; -i-y5/5! »- ., where y is any real number. These fetal” are the well-known expansions of cos y, sin y in powers "U 'ms' of the circular measure y. Where 2: is a complex number, the symbol e' may be defined to be such that its principal value is E(z); thus the principal values of ef#/, e"f/ are E(iy), E(-iy) The above expressions for cos y, sin y may then be written cos y=§ (ef"+e“f1'), sin y=§ 'i(ef1/-e"f”). These are known as the exponential values of the cosine and sine. It can be shown that the symbol ezas defined here satisfies the usual laws of combination for exponents.
22. The two functions cos z, sin z may be defined for all complex or real values of z by means of the equations eos y=§ {E(z) + E(-rz) }, sin 2 = (§){E(z) -E(-2) }, where E(z) represents the sum-function of 1 -Jr 2-l- z2, '2! -|-, + z"/nl + For real values of z this is in accordance with the ordinary definitions, as appears from the series obtained above for cos y, sm y. The fundamental properties of cos z, sin z can be deduced from this definition. Thus sin z = E(z), cos z -i sin z = E(-iz); therefore cos'zj+-sin2z=E(iz) E(-iz) =1. Again cos (z, -l-z2) is given by élE(12. + iz2) + E(- iz, -122) } =% {E(i21)E (wi) + E(7 121) E (-qz2)} Of ilE(1Z1)'l' E('“1Z1)}lE(¢Z2)'i'E<"1Z2)l'l'i E(¢Z1)"E(-'LZ1)l {E(zz2)-E(-122)}, whence we have cos (21-+122 = cos zl cos zzsin zl sin 22. Similarly, we find that sin (zl-4-z2)=sin z, cos 22+ cos z, sin zz. Again the equation E(z) = I has no real roots except
- =o, for e">1, if z is real and >o. Also E(z)=1 has no complex
root o.-HB, for o.-id would then also be a root, and E(2a) = E(a.+i8)E(a-iB) =I, which is impossible unless m=0. The roots of E(z) = I are therefore purely imaginary (except z=0); the smallest numerically we denote by 2 1l1r, so that E<2'iT)=I~ We have then E(2'i1rr)={E(2i'n')}'=I, if r is any integer; therefore 2i1rr is a root. It can be shown that no root lies between 2i1rr and 2(r+I)i1r; and thus that all the roots are given by z= =f=2i1rr. Since E(}'-l-217-rr)=E(z)E(2i1r)=E(z), we see that E(z), is periodic, of period 2'i7I'. It follows that cos z, sin z are periodic, of periods 21r The number here introduced may be identified with the ratio of the circumference to the diameter of a circle by considering the case of real values of z.
23. Consider the binomial theorem cos z+i
(LlTb)" = U"+nd" b+7E£%?l2 a"'”*b"+ . . . (n I) (+I)
W * n'-7 n r n
- nd gcijlnes + rl a by+' ' ' +I)
i" Serie' °f Putting a =eL9, b =e'|0, we obtain Sines and
cosines of (2 cos 6)"=2 cos m9-i-nz cos n-20 fgtlple +"-(Z I) 2 cos rl-{-. . +71—1—(n I)';'!(n r+I)2cos(n-2r)6+.. ”( I) ff"-l-3)
When n is odd the last term is eos 0, 2 ' ¢
and when n is even it is
5 .
If we put a=e, b= -e'»9, we obtain the formula (- i)%»(: sin0)"=2 cosn0-2ncos (n-2)0-l-'fl-IL? 2 cos (11-4)0- . +(- 1)"* 2 cos(n-2r)0.
+< I>gn<n~»>é#e»+1>
when n is even, and " A
(— 1)&( >(2 sin 0)" = 2sin m9-n 2 sin(n-2)0-|-$22 sin(n -4)0... +()'3?n(n-1). %(n-Q-3)
~' 9
5(1)-I)! ”'“
when n is odd. These formulae enable us to express any positive integral power of the sine or cosine in terms of sines or cosines of multiples of the argument. There are corresponding formulae when n is not a positive integer.
Consider the identity log(1-px)+log(1-qx)= log(I -p+qx-l-pgxz). Expand both sides of this equation in powers of x, and equate the coefficients of xn, we then get
12"-l-Q"' = (1>+q)"- n(1>+9)"'2l>q of (-3), , s
4% (P 'l'9) '1'292+- ° ° C2;'f:i;n:f +< I)'n(n-r-1)(n-VX2).. (n-21-1-1) Am (P“|'9)"'2'P'9'+- — If
we write this series in the reverse order, we have fl Il
i>'~+q"=2<- of <1>q>3-§ '§ <z>q>2"1(Q“i)2 2 2 2
+f'¥;! 3 (1>q>2
2 2 2 'L L1
f "f @(pq)2 3 G+. +(- 1)2 § (p+g)» when n is even, and
YL-I 1; i;
- 'E 1
+ (Pq) 2 "+. ..+<-1>"2;<1>+q>when n is odd. If in these three formulae we put P =e°9, q =e-'19, we obtain the following series for cos 110:- 2 eos n0= (2 eos 0)"-n(2 cos 9)” 2+ (2 cos 0)"'4-. +(I), n(n-r- 1)(n-152). (n-2r+I)(2 Cos 0>n 2' +' Q 1(7)
when n is any positive integer; 2 2 2 2 2(-I)2cos
u0= I-300520-i-n (n4, 22) c0s'0-fl (712 22 (nz 42) c0559 fl
+. +(-1)§ 2"“1 cos "0 (8)
when n is an even positive integer; n-1
(- I)T cos n0=n cos n6-n( ;, 12) cos"0+"(n2 12,02 32)cos56-1 +(-1)%" 2 "rl cos "0
(9)
when n is odd. If in the same three formulae we put p=e19, g= - e-19, we obtain the following four formulae:- fl
(- I)§ 2 eos n0= (2 sin 0)"-n(2 sin 0)"”'2-I-@ (2 sin 0)"'4n(n-r-1) (n 2r 1)
+(- I)' (2 Sin 0)"'”'+. . (1L even); (10) -1
(- I)%2 sin n6=the same series (n odd); (11) 2 'Z
cos nB;= I - sin?0 + sin40 - sin“0 +. . +2"“1 sin "6 (n even); (12) sin 116 = n sin 0 - n( 23T 12) sin30 'l' nw- - I;)in2 32) sin59 - . 1
+< o'%2»-1 Simo (n Odd). (13)
Next consider the identity f
<1 =, P q
I Px 1~Qx I”'(P+!l)x+PQx2
Expand both sides of this equation in powers of x, .and equate the coefficients of x"", then we obtain the equation
pn-gi = (1>+a>"-'- (11 - 2) <1> +q>'~-3 pa (-)(-)
- 7' 32 T* 4
(1>+q)""'1>'¢1'- - .<-1>f( )
<”"Q2> "'( ' 2” <p+q>"“"-1p'gf+ If, as before, we write this in the reverse order, we have the series < [»(fL;1) M* (ey) 3<pq>f”
+n<n2 22)y(nz 42
~ .95 If-
r—g *'l(f;g)5(1>Q)' +-..+(-I)z (1>+q)""
