Page 139 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 12.5 Summary

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.

Inverse Function Theorem

Let a function w = f(z) be analytic at a point z = z where f (z )  0 and w = f(z ).
0
0
0
0
Then there exists a neighbourhood of w in the w-plane in which the function w = f(z) has
0
a unique inverse z = F(w) in the sense that the function F is single-valued and analytic in
that neighbourhood such that F(w ) = z and
0
0
1
F(w) = .
f'(z)
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 1
Res (z = ) =  f(z) dz, | or Res (z = ) –  f(z) dz,
2 i C 2 i C


Integration taken in positive sense
the integration being taken round C in the negative sense w.r.t. the origin, provided that
this integral has a definite value. By means of the substitution z = w , the integral defining
-1
the residue at infinity takes the form

1 dw
1 [ f(w )] ,

2 i   w 2

taken in positive sense round a sufficiently small circle with centre at the origin.
(i) If the function f(z) has a simple pole at z = a, then, Res (z = a) = lim (z–a) f(z).

z a
(ii) If f(z) has a simple pole at z = a and f(z) is of the form f(z) =  (z) i.e. a rational
 (z)
function, then

Res (z = a) = lim (z-a) f(z) = lim (z-a)  (z)
z a z a  (z)

(z)

= lim a  (z)   (a)

z
z a

 (a)
= ,
 '(a)
where (a) = 0, (a)  0, since (z) has a simple zero at z = a

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