Page 133 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 1
= [exp (ima e i/4 ) + exp (ima e iz/4 )]
4





= 1   exp ima   i 1      exp ima    1 i     


4    2    2 
1   ma    ima    ima  
= exp  exp    exp 


4  2    2   2 
1   ma   ma 
= exp cos
2   2     2  
Hence by Cauchy’s residue theorem,
f(z) dz = f(z) dz + R R  f(x) dx = i exp    ma   cos   ma  


C T   2   2 
Taking limit as R and using (1), we get
x e
3 imx
   x  a 4 dx   i exp     ma   cos    ma  
4
2 
2 
Equating imaginary parts, we obtain
3
   x sinmx dx   exp     ma   cos    ma  
4
4
a
x 
2 
2 
3
or 0   x sinmx dx   exp     ma   cos    ma  
2
a
4
4
2 
x 
2 
12.4 Multivalued Function and its Branches
The familiar fact that sin q and cos q are periodic functions with period 2p, is responsible for the
non-uniqueness of  in the representation z = |z|e i.e. z = re . Here, we shall discuss
i
i
non-uniqueness problems with reference to the function arg z, log z and z . We know that a
a
function w = f(z) is multivalued when for given z, we may find more than one value of w. Thus,
a function f(z) is said to be single-valued if it satisfies
f(z) = f(z(r, )) = f(z(r,  + 2))
otherwise it is classified as multivalued function.
For analytic properties of a multivalued function, we consider domains in which these functions
are single valued. This leads to the concept of branches of such functions. Before discussing
branches of a many valued function, we give a brief account of the three functions arg z, log z
and z .
a
Argument Function

For each z  , z  0, we define the argument of z to be
arg z = [arg z] = {  R : z = |z|e }
i

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