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© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany

CAUCHY's theorem

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Let us consider the two-dimensional case. For  n=2  there are exactly two vectors of length  1 , which are orthogonal to  νΩ(x)  (see figure fig:tau). We set
τΩ(x):=i νΩ(x).
Here  R2  is identified with  C . Thus  τΩ(x)  is the vector  νΩ(x)  after a counterclockwise rotation by  90∘ , and  τΩ(x)∈Tx(∂Ω) . Then for  f=(f1+i f2):C → C 
f•τΩ = f•(i νΩ) =(-i f)•νΩ = (f2,-f1)•νΩ.
Let  f  be continuously differentiable. Then   div (f2,-f1) = ∂1f2-∂2f1 , hence the divergence theorem gives
[eq:6-10]
 
∫
∂Ω
 f•τΩ dH1 =
 
∫
∂Ω
 (f2,-f1)•νΩ dH1 =
 
∫
Ω
 div (f2,-f1) dL2 =
 
∫
Ω
 (∂1f2-∂2f1) dL2 .

-*- FIGURE NOT AVAILABLE -*-
Tangential- und Normalenvektor  [fig:tau]

Definition    [definition:6-9]
Let  Ω⊂C  be an open set. For  f∈C1(Ω;C)  the WIRTINGER derivatives are defined by
∂ conj(z)f
:=
1
2
 (∂1f+i∂2f) ,
∂zf
:=
1
2
 (∂1f-i∂2f) .
For these WIRTINGER derivatives the usual product rules hold, and it is
conj(∂zf)=∂ conj(z) conj(f)      and      conj(∂ conj(z)f) = ∂z conj(f) .
Example: For  f(z):=z  one has  ∂zf=1  and  ∂ conj(z)f=0 . For  g(z):= conj(z)  one has  ∂zg=0  and  ∂ conj(z)g=1 .

With these definitions the divergence theorem in two space dimensions becomes:

CAUCHY's theorem    [satz:6-10]
Let  Ω⊂R2  be a bounded GAUSS domain (see definition definition:6-7),  f∈C0(clos(Ω)) ∩C1(Ω;C)  with  ∂ conj(z)f ∈L1(Ω)  and  τΩ:=i νΩ . Then
 
∫
∂Ω
 fτΩ dH1 = 2i 
 
∫
Ω
 ∂ conj(z)f dL2 .
(Here the product  fτΩ  means the multiplication of complex numbers (!).)

Proof (Version 1). For simplicity write  τ:=τΩ  and  ν:=νΩ . Then, by definition of  τ ,
f τ= i f ν= (ν•e1)i f - (ν•e2) f .
Therefore the divergence theorem (in the version of sect:6-8-(3)) implies
 
∫
∂Ω
 fτdH1 =
 
∫
∂Ω
 (ν•e1) i f dH1 -
 
∫
∂Ω
 (ν•e2) f dH1 =
 
∫
Ω
 (∂1(i f)-∂2f) dL2 .
Since  ∂1(i f)-∂2f = 2i ∂ conj(z)f , this gives the assertion.

Proof (Version 2). For simplicity write  τ:=τΩ  and  ν:=νΩ . Writing  f=f1+i f2  one computes
[eq:6-10a]
f τ= (f1τ1-f2τ2) + i (f1τ2+f2τ1) = (f1,-f2)•τ+ i (f2,f1)•τ.
Applying eq:6-10 to the vector fields  (f1,-f2)  and  (f2,f1) , one obtains
 
∫
∂Ω
 fτdH1
=
 
∫
∂Ω
 (f1,-f2)•τdH1 +
 
∫
∂Ω
 (f2,f1)•τdH1
=
 
∫
Ω
 

((-∂1f2-∂2f1)+i (∂1f1-∂2f2))
=i ∂1f-∂2f = 2i ∂ conj(z)f
 dL2 .
This proves the assertion.

Alternatively, instead of eq:6-10a compute

[eq:6-10b]
f τ= - (f2,f1)•ν+ i (f1,-f2)•ν.
Hence the divergence theorem satz:6-7, applied to the vector fields  (f2,f1)  und  (f1,-f2) , directly gives
 
∫
∂Ω
 fτdH1
=
-
 
∫
∂Ω
 (f2,f1)•νdH1 +
 
∫
∂Ω
 (f1,-f2)•νdH1
=
 
∫
Ω
 (- div (f2,f1)+i  div (f1,-f2)) dL2 .

Remark: The integrals in satz:6-10 can also be written in terms of differential forms (differential forms will be introduced later (see sect:9-23):
 
∫
∂Ω
 fτΩ dH1
=
 
 
∫
(∂Ω,τΩ)
 f dz
2i 
 
∫
Ω
 f dL2
=
 
 
∫
(Ω,o+)
 ∂ conj(z)f d conj(z)∧dz .
Here  ∂Ω  is equipped with the orientation  τΩ , defined as above, and  o+  denotes the canonical orientation of  R2 .

In particular, CAUCHY's theorem is important for functions  f  satisfying  ∂ conj(z)f=0 :

Definition (Holomorphic functions)    [definition:6-11]
Let  Ω⊂R2  be an open set. A complex valued function  f∈C1(Ω;C)  is called holomorphic, if one of the following three equivalent properties is satified:
  • [definition:6-11-(i)]  f=f1+if2  satisfies the CAUCHY-RIEMANN differential equations
    ∂1f1
    =
    ∂2f2 ,
    ∂2f1
    =
    -∂1f2 .
  • [definition:6-11-(ii)]  ∂ conj(z)f=0 .
  • [definition:6-11-(iii)]  f  is complex differentiable in every point, that is for  z0∈Ω  there exists a complex number  f′(z0)∈C  with
    f(z)-f(z0)
     
    z-z0
     
    		 → f′(z0)     for z≠z0 as z → z0 .

Proof definition:6-11-(i)  <==>  definition:6-11-(ii). Let  f=f1+if2 . Then
∂ conj(z)f
=
1
2
 (∂1f+i∂2f)
=
1
2
 (∂1f1+i∂1f2+i∂2f1-∂2f2)
=
1
2
 ((∂1f1-∂2f2)+i(∂1f2+∂2f1)) .
Proof definition:6-11-(ii)  <==>  definition:6-11-(iii). For the derivative
Df(z0)(v)
=
v1∂1f(z0)+v2∂2f(z0)
=
v1(∂zf(z0)+∂ conj(z)f(z0)) -i v2(∂ conj(z)f(z0)-∂zf(z0))
=
∂zf(z0)v+∂ conj(z)f(z0) conj(v)
The definition of real differentiability is
f(z)-f(z0)=Df(z0)(z-z0)+|z-z0|ε(z) ,
where  ε(z) → 0  as  z → z0 . Hence
f(z)-f(z0)
 
z-z0
 
 = ∂zf(z0) + ∂ conj(z)f(z0)
conj(z-z0)
 
z-z0
 
 +ε̃(z)
with  ε̃(z) → 0  as  z → z0 . Thus the left-hand side has a limit for  z → z0 , if and only if  ∂ conj(z)f(z0)=0 .

Version 1.7
H.W. Alt - 02.01.2007

GREEN's formula GREEN's formula
Partial integration in  Rn  Partial integration in  Rn 
Isolated singularities Isolated singularities
Isolated singularities Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany