Page 138 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 12: Fundamental Theorem of Algebra

The function in (6) has infinitely many different values. But the number of different values of z a Notes
will be finite in the cases in which only a finite number of the values e 2 ian , nI, are different from
one another. In such a case, there must exist two integers m and m (m = m) such that e 2 iam =
e 2 iam or e 2 ia(m -m) = 1. Since e = 1 only if z = 2in, thus we get a (m-m) = n and therefore it follows
z
that a is a rational number. Thus, z has a finite set of values iff a is a rational number. If a is not
a
rational, z has infinity of values.
a
We have observed that if z = re and a is any real number, then the branch
i
log z = log r + i (r > 0,  <  <  + 2) (7)
of the logarithmic function is single-valued and analytic in the indicated domain. When this
branch is used, it follows that the function (5) is single valued and analytic in the said domain.
The derivative of such a branch is obtained as

d (z ) = d a
a
dz dz [exp (a log z)] = exp ( a log z) z

exp(alog z)
= a = a exp [(a -1) log z]
exp(log z)
= az .
a-1
As a particular case, we consider the multivalued function f(z) = z and we define
1/2
z = r e i/2 , r > 0,  <  <  + 2 (8)
1/2
where the component functions

u(r, ) = r cos /2, v(r, ) = r sin /2 (9)
are single valued and continuous in the indicated domain. The function is not continuous on the
line  =  as there are points arbitrarily close to z at which the values of v (r, ) are nearer to
r sin /2 and also points such that the values of v(r, ) are nearer to – r sin /2. The function
(8) is differentiable as C-R equations in polar form are satisfied by the functions in (9) and
d (z 1/2 i   i  1 1 




dz ) = e (u  iv ) e   2 r cos /2 i 2 r sin /2  
r
r
1 1
= e  i /2 

2 r 2z 1/2
Thus, (8) is a branch of the function f(z) = z where the origin and the line  =  form branch cut.
1/2
When moving from any point z = re about the origin, one complete circuit to reach again, at z,
i
we have changed arg z by 2. For original position z = re , we have w = r e i/2 , and after one
i
complete circuit, w = r e i(+2)/2 = - r e i/2 . Thus, w has not returned to its original value and
hence, change in branch has occurred. Since a complete circuit about z = 0 changed the branch of
the function, z = 0 is a branch point for the function z .
1/2

LOVELY PROFESSIONAL UNIVERSITY 131