§2. Series expansions
373
Put e(x) = c(x) + is(x). The addition formulae show, by a trivial
calculation, not even assuming that c(x) and s(x) are real-valued, that
e(x + y) = e(x)e(y). Moreover, the function e(x) has derivative e'(O) = i
at x = 0 by hypothesis (ii). Corollary 1 of Theorem 11 immediately finishes
the proof, the series (3') and (3") being obtained by separating the real and
imaginary parts of the exponential series.
Theorem 12 leads us to define the functions cos x and sin x by the formulae (3), for x complex as well. We then obtain, as we shall see, all the
elementary properties of the trigonometric functions.
(i) Numerical calculation. The series cos x and sin x converge with great
rapidity. If one calculates exp( z) from its nth partial sum, the error committed
is less than
Izl[n] (1 + Izl/(n + 1) + IzI2/(n + l)(n + 2) + ... +) ~ Izln. exp(lzl)/n!
with a bound of the same kind for the trigonometric parallels. For reasonable values of Izl - for the trigonometric functions, it is enough to do the
calculations for 0 < z < 7r/2 < 1.6 -, the denominator n!, about 4.10 7 for
n = 11, the sixth term of the series sinx, rapidly becomes enormous with
respect to the factor exp(lzl)lzln, so that the approximation is excellent even
for moderate values of n.
(ii) Euler's Relations. As we have already seen, the series expansions (3)
trivially imply the relations
exp(iz) = cos z + i sin z,
exp( -iz) = cos z - i sin z.
From these we deduce that
(14.4)
sin z = (e iZ - e- iZ ) /2i,
the famous and fundamental formulae which frequently facilitate trigonometric calculations.
(iii) Addition formulae. For x and y real, we may write
exp(ix) exp(iy) = exp(i(x + y))
and separate the real and imaginary parts using (2). Immediately we obtain
(14.5')
(14.5")
cos(x + y)
sin(x + y)
cos x cosy - sinxsiny,
sin x cosy + cosxsiny.
For x and y complex, we have to use the formulae (4) and write a few lines
of the kind (a + b)(c + d) = etc.