Page 140 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 12: Fundamental Theorem of Algebra

(iii) If f(z) has a pole of order m at z = a then we can write Notes

 (z)
f(z) =
(z a) m

where (z) is analytic and (a)  0.

1 1  (z)
Now, Res (z = a) = b = f(z) dz = dz

1 2 i  C 2 i (z a) m



C
1 | m 1  (z)

=  m 1 1 dz

| m 1 2 i C (z a)  


1
=  (a) [By Cauchy’s integral formula for derivatives]
m-1
| m 1

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.
12.6 Keywords
n zeros: Every polynomial of degree n has exactly n zeros.

Inverse function: If f(z) = w has a solution z = F(w), then we may write f{F(w)} = w, F{f(z)} = z. The
function F defined in this way, is called inverse function of f.
Residue at infinity: If f(z) is analytic or has an isolated singularity at infinity and if C is a circle
enclosing all its singularities in the finite parts of the z-plane, the residue of f(z) at infinity is
defined by

1
Res (z = ) =  f(z) dz,
2 i C

Cauchy Residue Theorem: Let f(z) be one-valued and analytic inside and on a simple closed
contour C, except for a finite number of poles within C. Then

 f(z) dz = 2i [Sum of residues of f(z) at its poles within C]
C
Multivalued function: 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.

12.7 Self Assessment

1. Every polynomial of degree n has exactly ................
2. If f(z) = w has a solution z = F(w), then we may write f{F(w) } = w, F{ f(z)} = z. The function
F defined in this way, is called ................

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