Complex Analysis - Maths KU

Recall: Branches of a multiple-valued
function F are denoted by f1, f2
and so on.
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4.2 Complex Powers 197
Definition 4.1 in Section 4.1 gives
e −1+i(loge 3)/π = e −1
�
cos loge 3
π + i sin log �
e 3
,
π
and so, (−3) i/π ≈ 0.3456 + 0.1260i.
(b) For z =2i, we have |z| = 2 and Arg(z) =π/2, and so Ln 2i = log e 2+iπ/2
by (14) in Section 4.1. By identifying z =2i and α =1− i in (6) we
obtain:
(2i) 1−i = e (1−i)Ln2i = e (1−i)(log e 2+iπ/2) ,
or (2i) 1−i = e log e 2+π/2−i(log e 2−π/2) .
We approximate thisvalue using Definition 4.1 in Section 4.1:
(2i) 1−i = e loge 2+π/2�
�
cos loge 2 − π
� �
− i sin loge 2 −
2
π
��
2
≈ 6.1474 + 7.4008i.
Analyticity In general, the principal value of a complex power z α defined
by (6) isnot a continuousfunction on the complex plane because the
function Ln z isnot continuouson the complex plane. However, since the
function e αz iscontinuouson the entire complex plane, and since the function
Ln z iscontinuouson the domain |z| > 0, −π