Page 134 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 12: Fundamental Theorem of Algebra

the square bracket notation emphasizes that arg z is a set of numbers and not a single number. Notes
i.e. [arg z] is multivalued. In fact, it is an infinite set of the form { + 2n : n  I}, where  is any
z
fixed number such that e = .
i
|z|
For example, arg i = {(4n +1) /2 : n  I}

1 
Also, arg    = {– :   arg z}
 z 
Thus, for z , z  0, we have
2
1
arg (z z ) = { +  :   arg z ,   arg z }
2
2
1
1
1
2
2
1
= arg z + arg z 2
1
and arg = arg z – arg z 2
1
For principal value determination, we can use Arg z = , where z = |z| e , – <    (or 0  
i
< 2). When z performs a complete anticlockwise circuit round the unit circle,  increases by 2
and a jump discontinuity in Arg z is inevitable. Thus, we cannot impose a restriction which
determines  uniquely and therefore for general purpose, we use more complicated notation
arg z or [arg z] which allows z to move freely about the origin with  varying continuously. We
observe that
arg z = [arg z] = Arg z + 2n, n  I.
Logarithmic Function
We observe that the exponential function e is a periodic function with a purely imaginary
z
period of 2i, since
e z+2i = e . e = e , e = 1.
z
2i
z
2i
i.e. exp (z + 2i) = exp z for all z.
If w is any given non-zero point in the w-plane then there is an infinite number of points in the
z-plane such that the equation
w = e z (1)

is satisfied. For this, we note that when z and w are written as z = x + iy and w =  e (- <   ),
i
equation (1) can be put as
e = e x+iy = e e = r e i (2)
iy
z
x
From here, e =  and y =  + 2n, n  I.
x
Since the equation e =  is the same as x = log  = log  (base e understood), it follows that when
x
e
w =  e (- <   ), equation (1) is satisfied if and only if z has one of the values
i
z = log  + i ( + 2n), n  I (3)
Thus, if we write
log w = log  + i ( + 2n), n  I (4)

we see that exp (log w) = w, this motivates the following definition of the (multivalued)
logarithmic function of a complex variable.

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