Page 135 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes The logarithmic function is defined at non-zero points z = re (- <   ) in the z-plane as
i
log z = log r + i ( + 2n), n  I (5)

The principal value of log z is the value obtained from (5) when n = 0 and is denoted by Log z.
Thus,
Log z = log r + i i.e. Log z = log |z| + i Arg z (6)

Also, from (5) & (6), we note that
log z = Log z + 2ni, n  I (7)
The function Log z is evidently well defined and single-valued when z  0.
Equation (5) can also be put as

log z = {log |z| + i :   arg z}
or [log z] = {log |z| + i :   [arg z]} (8)
or log z = log |z| + i = log |z| + i arg z (9)
where  =  + 2np,  = Arg z.

From (8), we find that
log 1 = {2ni, n  I}, log (–1) = {(2n+1) pi, n  I}
In particular, Log 1 = 0, Log (–1) = i. Similarly log, log i = {(un+1) i/2, nI}, log (–i) = {un-1)
i/2, nI} In particular, Log i = i/2, Log (–i) = –i/2.
Thus, we conclude that complex logarithm is not a bona fide function, but a multifunction. We
have assigned to each z  0 infinitely many values of the logarithm.

Complex Exponents

When z  0 and the exponent a is any complex number, the function z is defined by the equation.
a
w = z = e a log z = exp (a log z) (1)
a
where log z denotes the multivalued logarithmic function. Equation (1) can also be expressed as
w = z = {e a(log |z| + i) :   arg z}
a
or [z ] = {e a(log |z| + i) :   [arg z]}
a
Thus, the multivalued nature of the function log z will generally result in the many-valuedness
of z . Only when a is an integer, z does not produce multiple values. In this case, z contains a
a
a
a
1
single point z . When a = (n = 2, 3, ), then
n
n
w = z = (r e ) = r e i( + 2m )/n , m  I
1/n
1/n
i 1/n
We note that in particular, the complex nth roots of ±1 are obtained as
w = 1  w = e 2mi/n , w = –1  w = e (2m+1)i/n , m = 0,1, , n-1.
n
n
For example, i = exp (–2 i log i) = exp [–2i (4n+1) i/2]
-2i
= exp [(4n+1) ], n  I
It should be observed that the formula
x x = x , x, a, b,  R
a+b
a
b

128 LOVELY PROFESSIONAL UNIVERSITY