1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 57

zn has just one value, which coincides with the one defined in Section 1.11;

and when w is an irreducible fraction m/n (n ~ 2), the power zm/n has

exactly n values, which coincide with those given by Definition 1.7.
In fact, for w = n we have
Zw = enlogz = en[lnr+i(0+2k11')]
= en In r+in0+2nk1l'i = en In r+inO

= rn( cos nB + i sin nB)

which is precisely the value given by (1.11-3). For w = m/n we have

zw = e<mfn)logz = e<mfn)lnr+im0/n+2mk1l'i/n

= rmfnei(m0+2mk11')/n

which is the same as expression (1.13-2). If w = 'Y is an irrational number,

we have

The difference between two exponents corresponding to k = k 1 and

k = k 2 (k 1 f k 2 ) is 2"f(k 1 - k 2 )'1ri, which is not an integral multiple of 27ri

(because of the irrational factor 'Y)· Therefore, the power zw a~sumes in
this case denumerably many values.
Similarly, there are denumerably many values when w is imaginary.

Example

(-1r = ei7l'(7l'+^2 k7l') = cos(2k + l)7r^2 + i sin(2k + l)7r^2

(k = 0, ±1, ±2, ... )

Letting w = u + iv, we have

Zw = e(u+iv)[ln r+i(IJ+2k11')]

= euln r-v(0+2k11') • ei[v In r+u(0+2k11')]

Hence
lzwl = eulnr-v(0+2k11')

argzw = vlnr + u(B + 2h) + 2n7r

where k and n are arbitrary integers. This shows that when v f 0 both

the modulus and the argument of zw are multiple-valued. If v = 0, then

w = u, and

\zwl = eulnr = ru = \z\w

However, when v f 0 we have seen that

lzwl = rue-v(e+2k11') = lzlue-v(e+2k11')