Complex Analysis - Maths KU

Note
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2.4 Special Power Functions 87
If we set θ =Arg(z) and replace z with re iθ in (7), then we obtain an
alternative description of the principal square root function for |z| > 0:
z 1/2 = √ re iθ/2 , θ = Arg(z). (8)
You should take note that the symbol z 1/2 used in Definition 2.4 represents
something different from the same symbol used in Section 1.4. In (7) we
use z 1/2 to represent the value of the principal square root of the complex
number z, whereas in Section 1.4 the symbol z 1/2 was used to represent the
set of two square roots of the complex number z. This repetition of notation is
unfortunate, but widelyused. For the most part, the context in which you see
the symbol z 1/2 should make it clear whether we are referring to the principal
square root or the set of square roots. In order to avoid confusion we will, at
times, also explicitlystate “the principal square root function z 1/2 ” or “the
function f(z) =z 1/2 given by(7).”
EXAMPLE 5 Values of z1/2 Find the values of the principal square root function z1/2 at the following
points:
(a) z =4 (b) z = −2i (c) z = −1+i
Solution In each part we use (7) to determine the value of z 1/2 .
(a) For z = 4, we have |z| = |4| = 4 and Arg(z) =Arg(4) = 0, and so from
(7) we obtain:
4 1/2 = √ 4e i(0/2) =2e i(0) =2.
(b) For z = −2i, we have |z| = |−2i| = 2 and Arg(z) =Arg(−2i) =−π/2,
and so from (7) we obtain:
( −2i) 1/2 = √ 2e i(−π/2)/2 = √ 2e −iπ/4 =1− i.
(c) For z = −1+i, wehave|z| = |−1+i| = √ 2 and Arg(z) =Arg(−1+i) =
3π/4, and so from (7) we obtain:
(−1+i) 1/2 ��√ �
= 2 e i(3π/4)/2 = 4√ 2e i(3π/8) ≈ 0.4551 + 1.0987i.
It is important that we use the principal argument when we evaluate the
principal square root function of Definition 2.4. Using a different choice for
the argument of z can give a different function. For example, in Section 1.4
we saw that the two square roots of i are 1
√ √ √ √
1
1 1
2 2+ 2 2 i and − 2 2 − 2 2 i.
For z = i we have that |z| = 1 and Arg(z) =π/2. It follows from (7) that:
i 1/2 = √ 1e i(π/2)/2 =1· e iπ/4 =
√ 2
2 +
√ 2
2 i.