§2. Series expansions
377
or v = - sint = sin( -t). On replacing t by -t if need be, we see that we can
always put z in the form
(14.17)
z=r(cost+isint) = Izlexp(it).
The number t is unique up to a multiple of 27r. If indeed t + e is another
solution, we have exp(it) = exp(it + ic), whence exp(ic) = 1, and then
exp(i(x + e)) = exp(ix) for any x E JR, and e is a period of the trigonometric
functions, so of the form 2k7r as we saw above. "The" number t is called the
argument of the complex number z -=I o. We shall study it in more detail in
§ 4 of this Chapter.
As one can always put r in the form exp s with s E JR, and in fact s = log r,
the relation (17) can again be rewritten, as z = exp(s + it). In other words,
the exponential function maps C onto C* = C - {O}. But in contrast to what
happens in the real domain, the map is not injective, as we have just seen.
(ix) Periodicity of exp z. The calculation of the values of cos x and sinx
for x = 7r or x = 27r shows that
(14.18)
e i7r = exp(i7r) = -1,
e 2i7r = exp(2i7r) = 1,
whence more generally
(14.19)
exp(z+k7ri) = (-I)k expz .
The function exp z thus has complex periods, namely all the multiples of
27ri, and in particular is no more injective than the trigonometric functions.
It has no other periods: the relation exp z = 1, with z = x + iy, implies
I expzl = expx = 1, whence x = 0, and exp(iy) = 1, Le. cosy = 1 and
sin y = 0, whence y = 2k7r as we showed above. In other words,
exp z = 1 <===} z = 2ki7r.
(x) Logarithms of a complex number. One defines "the" logarithm of a
complex number z -=I 0 by agreeing that
u = .cog(z) <===} z = expu,
but the pretend function .cog is not a function: to each value of z there
correspond infinitely many values of u, differing one from the other by a
multiple of 2i7r. This is a difficulty of the same kind as arises in defining
"functions" such as arcsin x in the real domain. In other words, handle with
caution. More exactly, put z = Izl exp(it) with t = arg(z). Then
z = exp(log Izl + it)
377
or v = - sint = sin( -t). On replacing t by -t if need be, we see that we can
always put z in the form
(14.17)
z=r(cost+isint) = Izlexp(it).
The number t is unique up to a multiple of 27r. If indeed t + e is another
solution, we have exp(it) = exp(it + ic), whence exp(ic) = 1, and then
exp(i(x + e)) = exp(ix) for any x E JR, and e is a period of the trigonometric
functions, so of the form 2k7r as we saw above. "The" number t is called the
argument of the complex number z -=I o. We shall study it in more detail in
§ 4 of this Chapter.
As one can always put r in the form exp s with s E JR, and in fact s = log r,
the relation (17) can again be rewritten, as z = exp(s + it). In other words,
the exponential function maps C onto C* = C - {O}. But in contrast to what
happens in the real domain, the map is not injective, as we have just seen.
(ix) Periodicity of exp z. The calculation of the values of cos x and sinx
for x = 7r or x = 27r shows that
(14.18)
e i7r = exp(i7r) = -1,
e 2i7r = exp(2i7r) = 1,
whence more generally
(14.19)
exp(z+k7ri) = (-I)k expz .
The function exp z thus has complex periods, namely all the multiples of
27ri, and in particular is no more injective than the trigonometric functions.
It has no other periods: the relation exp z = 1, with z = x + iy, implies
I expzl = expx = 1, whence x = 0, and exp(iy) = 1, Le. cosy = 1 and
sin y = 0, whence y = 2k7r as we showed above. In other words,
exp z = 1 <===} z = 2ki7r.
(x) Logarithms of a complex number. One defines "the" logarithm of a
complex number z -=I 0 by agreeing that
u = .cog(z) <===} z = expu,
but the pretend function .cog is not a function: to each value of z there
correspond infinitely many values of u, differing one from the other by a
multiple of 2i7r. This is a difficulty of the same kind as arises in defining
"functions" such as arcsin x in the real domain. In other words, handle with
caution. More exactly, put z = Izl exp(it) with t = arg(z). Then
z = exp(log Izl + it)