Cauchy integral theorem

Cauchy integral theorem

Theorem 1

Let

U

⊂

C

be an open, simply connected domain, and let

f

:

U

→

C

be a

function

whose

complex derivative

, that is

lim

w

→

z

⁡

f

⁢

(

w

)

-

f

⁢

(

z

)

w

-

z

,

exists for all

z

∈

U

.
Then, the

integral

(

http://planetmath.org/Integral2

) around every closed

contour

γ

⊂

U

vanishes

; in symbols

∮

γ

f

⁢

(

z

)

⁢

𝑑

z

=

0

.

We also have the following, technically important

generalization

involving

removable singularities

.

Theorem 2

Let

U

⊂

C

be an open, simply connected domain, and

S

⊂

U

a finite subset. Let

f

:

U

\

S

→

C

be a function whose complex derivative exists for all

z

∈

U

\

S

, and that is

bounded

near all

z

∈

S

. Let

γ

⊂

U

\

S

be a closed contour
that avoids the exceptional points. Then, the integral of

f

around

γ

vanishes.

Cauchy’s

theorem

is an essential stepping stone in the theory of

complex analysis

. It is required for the proof of the

Cauchy integral
formula

, which in turn is required for the proof that the

existence

of
a complex derivative implies a

power series

representation

.

The original version of the theorem, as stated by Cauchy in the early
1800s, requires that the

derivative

f

′

⁢

(

z

)

exist and be

continuous

.
The existence of

f

′

⁢

(

z

)

implies the

Cauchy-Riemann equations

, which
in turn can be restated as the fact that the complex-valued

differential

f

⁢

(

z

)

⁢

d

⁢

z

is closed. The original proof makes use of
this fact, and calls on Green’s Theorem to conclude that the contour
integral vanishes. The proof of Green’s theorem, however, involves an
interchange of order in a

double integral

, and this can only be
justified if the integrand, which involves the real and

imaginary
parts

of

f

′

⁢

(

z

)

, is assumed to be continuous. To this date, many
authors prove the theorem this way, but erroneously fail to mention
the

continuity

assumption

.

In the latter part of the

19

th

century E. Goursat found a proof
of the integral theorem that merely required that

f

′

⁢

(

z

)

exist.
Continuity of the derivative, as well as the existence of all higher
derivatives, then follows as a

consequence

of the Cauchy integral
formula. Not only is Goursat’s version a sharper result, but it is
also more

elementary

and self-contained, in that sense that it is does
not require Green’s theorem. Goursat’s

argument

makes use of
rectangular contour (many authors use

triangles

though), but the

extension

to an arbitrary simply-connected domain is relatively
straight-forward.

Theorem 3

(Goursat)

Let

U

be an open domain containing a

rectangle

R

=

{

x

+

i

⁢

y

∈

ℂ

:

a

≤

x

≤

b

,

c

≤

y

≤

d

}

.

If the complex derivative of a function

f

:

U

→

C

exists
at all points of

U

, then the contour integral of

f

around the
boundary of

R

vanishes; in symbols

∮

∂

⁡

R

f

⁢

(

z

)

⁢

𝑑

z

=

0

.

Bibliography.

•

Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.

Title

Cauchy integral theorem

Canonical name

CauchyIntegralTheorem

Date of creation

2013-03-22 12:54:12

Last modified on

2013-03-22 12:54:12

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

16

Author

rmilson (146)

Entry type

Theorem

Classification

msc 30E20

Synonym

fundamental theorem of complex analysis

Related topic

ClosedCurveTheorem

Related topic

CauchyResidueTheorem

Defines

Goursat’s Theorem

Generated on Fri Feb 9 20:19:58 2018 by

LaTeXML