Page 137 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes  + 2. The origin and the ray  =  make up the branch cut for the branch (2) of the logarithmic
function. The function
Log z = log r + i (r > 0, – <  < ) (4)
is called the principal branch of the logarithmic function in which the branch cut consists of the
origin and the ray  = . The origin is evidently a branch point of the logarithmic function.

y


O x

For analyticity of (2), we observe that the first order partial derivatives of u and v are continuous
and satisfy the polar form

1 1
u = v , v   r u 
r
r
r

of the C-R equations. Further,
d (logz) = e (u + iv )
-i
dz r r


-i 
= e  1  i0  1

 r  re i
d 1
Thus, (logz) = (|z| = r > 0, < arg z < a + 2p)
dz z
In particular,

d (logz) = 1
dz z (|z| > 0, – < Arg z < ).

1
Further, since log = –log z,  is also a branch point of log z. Thus, a cut along any half-line
z
from 0 to  will serve as a branch cut.

Now, let us consider the function w = z in which a is an arbitrary complex number. We can write
a
w = z == e a log z (5)
a
where many-valued nature of log z results is many-valuedness of z . If Log z denotes a definite
a
branch, say the principal value of log z, then the various values of z will be of the form
a
e
z = e a(Log z + 2ni) = e a Log z 2 i a n (6)
a
where log z = Log z + 2ni, n  I.

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