Complex Analysis - Maths KU

186 Chapter 4 Elementary Functions
Observe that each of the values found in parts (a)–(c) could have also
been found by setting n = 0 in the expressions for ln z from Example 3.
Note: ☞ It isimportant to note that the identitiesfor the complex logarithm in
Theorem 4.3 are not necessarily satisfied by the principal value of the complex
logarithm. For example, it is not true that Ln(z1z2) =Ln z1 +Lnz2 for
all complex numbers z1 and z2 (although it may be true for some complex
numbers). See Problem 55 in Exercises 4.1.
Ln z as an Inverse Function Because Ln z is one of the valuesof
the complex logarithm ln z, it followsfrom (10) that:
e Lnz = z for all z �= 0. (16)
This suggests that the logarithmic function Ln z isan inverse function of
exponential function e z . Because the complex exponential function is not
one-to-one on itsdomain, thisstatement isnot completely accurate. Rather,
the relationship between these functions is similar to the relationship between
the squaring function z 2 and the principal square root function z 1/2 =
� |z|e iArg(z)/2 defined by (7) in Section 2.4. The exponential function must
first be restricted to a domain on which it is one-to-one in order to have a
well-defined inverse function. In Problem 52 in Exercises 4.1, you will be
asked to show that e z isa one-to-one function on the fundamental region
−∞