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IV - Powers, Exponentials, Logarithms, Trigonometric Functions
(iv) Parity. The formulae
cos(-x) = cosx,
sin( -x) = - sin x
follow from the power series (3') and (3"). It is even more obvious that
cosO = 1,
sinO=O.
(v) The Pythagorean Relation
(14.6)
cos 2 Z + sin 2 z = 1.
Even for z complex, the left hand side is equal to
(cos z + i sin z)(cos z - i sin z) = exp(iz) exp( -iz) = exp(O) = 1.
For z = x + iy with x, y E JR, we have e Z = eXe iy = eX(cosy + i siny), and
since I cosy + isinyl = 1 for y E JR, by (6), we conclude that
le z I = eRe(z)
and more generally that
(14.7)
for a > 0, z E since a Z = e Z log a with log a real; this formula is in constant use.
Corollary: the series «s) = E l/n s converges in the half plane Re(s) > 1,
which completes Theorem 5 of Chap. II, nO 12. Its sum is holomorphic in this
half plane since the series of derivatives
('(s) = - 2)og(n)/n s
converges normally on Re(s) ;:::: a for any a > 1 (see Chap. III, nO 17, example 3, the arguments extending immediately to the case where s is complex).
The same results for multiple series such as E(m 2 + n 2 )-s/2.
(vi) Derivatives. We have
(14.8)
cos' z = - sinz,
sin' z = cos z.
Simplest is to differentiate the power series (3') and (3") by straightforwardly
applying the general rule of Chap. II, nO 19. We can iterate the formulae (8) and find the successive derivatives, for example cos" z = - cos z,
sin'" z = - cos z, etc. These formulae are a fortiori valid in JR.