corollary of Cauchy integral theorem

corollary of Cauchy integral theorem

Theorem

.

Let

γ

be a closed

contour

of

ℂ

not intersecting itself and

γ

1

,

…

,

γ

n

likewise.  Let all the contours

γ

1

,

…

,

γ

n

be outside each other but insinde

γ

.  If the closed, “holey” domain between

γ

and the

γ

j

s is contained in a domain

U

where

f

is

holomorphic

, then

∮

γ

f

⁢

(

z

)

⁢

𝑑

z

=

∮

γ

1

f

⁢

(

z

)

⁢

𝑑

z

+

…

+

∮

γ

n

f

⁢

(

z

)

⁢

𝑑

z

(1)

where all

integrals

are taken with the same direction of circulation.  Especially, in the case

n

=

1

one has

∮

γ

f

⁢

(

z

)

⁢

𝑑

z

=

∮

γ

1

f

⁢

(

z

)

⁢

𝑑

z

.

(2)

Note 1.

The integrals in (1) and (2) need not necessarily vanish, since inside a

γ

j

there may be points not belonging to

U

.

Note 2.

When

n

=

0

,  i.e. the sum on the right hand side of (1) is empty (

http://planetmath.org/EmptySum

), it has the value 0; thus also the

Cauchy integral theorem

is a special case of (1).

Note 3.

The theorem implies easily the

residue theorem

.

Proof.

We prove the theorem only in the case

n

=

1

.  Other cases may be handled analogously.

Draw two auxiliary ways connecting

γ

and

γ

1

.  The integral of

f

taken anticlockwise around the route consisting of the upper parts of the curves and the auxiliary ways is, by the fundamental theorem of complex analysis, equal zero.  Similarly the integral of

f

taken anticlockwise around the route consisting of the lower parts of the curves and the auxiliary ways is equal zero.  Thus also the sum of both equals zero.  But in the sum, the portions taken along the auxiliary ways are run in opposite directions and so they cancel each other.  Therefore in the sum only the portions, which are run along

γ

and

γ

1

, remain; then

∮

γ

f

⁢

(

z

)

⁢

𝑑

z

+

∮

γ

1

f

⁢

(

z

)

⁢

𝑑

z

=

0

,

i.e.

∮

γ

f

⁢

(

z

)

⁢

𝑑

z

=

-

∮

γ

1

f

⁢

(

z

)

⁢

𝑑

z

.

However, here

γ

is run anticlockwise and

γ

1

clockwise.  Reversing the direction in the right hand side of this last equation, one obtains (2). Q.E.D.

Example.

Calculate

∮

C

d

⁢

z

z

-

z

0

where the circle

C

of

complex plane

with centre

z

0

and radius

R

is run once anticlockwise.

Since

|

z

-

z

0

|

=

R

we can take the

parametric presentation

z

-

z

0

=

R

⁢

e

i

⁢

φ

with

0

≦

φ

<

2

⁢

π

.

Then

d

⁢

z

=

i

⁢

R

⁢

e

i

⁢

φ

⁢

d

⁢

φ

and

∮

C

d

⁢

z

z

-

z

0

=

∫

0

2

⁢

π

i

⁢

𝑑

φ

=

2

⁢

i

⁢

π

.

(3)

By the equation (2) of the theorem, one can infer that the result (3) is true for any

continuous

contour going once around the point

z

0

anticlockwise (cf. the lemma of

http://planetmath.org/node/3105

this entry).

Title

corollary of Cauchy integral theorem

Canonical

name

CorollaryOfCauchyIntegralTheorem

Date of creation

2013-03-22 18:54:12

Last modified on

2013-03-22 18:54:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E20

Synonym

generalisation of Cauchy integral theorem

Related topic

VariantOfCauchyIntegralFormula

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