Complex Numbers 55
where ln r indicates, as usual, the natural or N aperian logarithm of the
positive real number r. Thus the set of values satisfying (1.15-1) is
{lnr+i(0+2br): r = lzl,O = Argz} (1.15-2)
Denoting by log z any value in this set, we usually write
w =log z = lnr + i(O + 2br), k = O, ±1, ±2,... (1.15-3)
'-- --. ---
Example ~- log(l + i) = ln v'2 + i ( ~ + 2br)
Since argz = 0 + 2k7r, we may also write (1.15-3) in the form
log z = ln lzl + i arg z (1.15-4)
· If we choose for arg z its principal value Arg z, the corresponding value
in (1.15-4) is called the principal value of the logarithm of z, denoted Log z.
Hence
Log z = ln z + i Arg z
Examples
1. Log(l+i) = lnv'2+i7r/4
- Log(-3) = ln3 + i7r
Replacing w by logz in (1.15-1), we have the identity
elog z = z
(1.15-5)
(1.15-6)
The properties of complex logarithms are similar to the corresponding
properties of real logarithms, as Theorem 1.12 shows.
Theorem 1.12 The complex logarithm has the following properties:
- log 1 = 2bri, Log 1 = 0
- loge = 1 + 2bri, Loge = 1
3. log(zz') = logz+logz' (mod27ri), z,z' /:- 0
4. log(z/z') = logz - logz' (mod27ri), z,z' /:- 0
- {logzn} ~ {nlogz}, n any integer, z /:- 0
provided that properties 3 and 4 are understood in the following sense:
adding, or subtracting, to a value of log z a value of log z', we obtain one
of the values of log(zz') or of log(z/ z'), respectively.
Property 5 means that the set of values of log zn contains the set of
values of n log z. The inclusion is proper except for n = +l.
Proofs Properties 1 and 2 follow at once from (1.15-4) and (1.15-5). To
prove properties 3 and 4 it suffices to write, using (1.15-6),
elog z = z and elog z' = z'
