analytic continuation

analytic continuation

Suppose that

D

1

and

D

2

are connected open subsets of the

complex plane

and that

D

1

⊂

D

2

. Suppose that

f

:

D

1

→

ℂ

and

g

:

D

2

→

ℂ

are

analytic

. If the

restriction

of

g

to

D

1

equals

f

, we say that

g

is a

direct analytic continuation

of

f

.

The reason that the notion of

analytic continuation

is interesting is the

rigidity theorem

for

complex functions

, which implies that analytic continuation is unique. This is in marked contrast to what would happen if, instead of analyticity, we only required a weaker condition such as

continuity

. In that case, there would exist a great number of

continuations

of a given

function

to a larger domain. In fact there are so many possibilities that the choice of continuation is largely arbitrary — one can find a continuation which agrees with any

continuous function

on a subset of

D

2

which is separated from

D

1

.

One can generalize this notion to continuation along a chain of open sets.
Suppose that one has a

sequence

of open sets

D

1

,

…

,

D

n

such that

D

k

∩

D

k

+

1

is not empty for

k

between

1

and

n

-

1

and

n

functions

f

k

:

D

k

→

ℂ

such that

f

k

⁢

(

z

)

=

f

k

+

1

⁢

(

z

)

when

z

∈

D

k

∩

D

k

+

1

for all

k

between

1

and

n

-

1

. Then we say
that

f

n

is the

indirect analytic continuation

of

f

1

along the chain

D

1

,

…

,

D

n

.

A related notion is that of analytic continuation along an arc. Given an arc

C

with

endpoints

x

and

y

, a function

f

defined in open neighborhood of

x

, and a function

g

defined in open neighborhood of

y

we say that

g

is the analytic continuation of

f

along the arc

C

if there exists an open set containing

C

and and an

analytic function

h

defined on this open set such that

f

agrees with

h

at those points of the complex plane where both

f

and

h

are defined and

g

agrees with

h

at those points of the complex plane where both

g

and

h

are defined. By the rigidity theorem, the analytic continuation of a function along an arc is unique.  This concerns especially extending a real function to an analytic function of a complex variable.

It is worth noting that it is possible to obtain different analytic continuations of the same function along two arcs with the same endpoints. For

instance

, if we let

f

be the square root function, let

x

=

1

and

y

=

-

1

, and let

C

1

and

C

2

be the two halves of the

unit circle

joining

x

to

y

, then the result of analytically continuing

f

from

x

to

y

along

C

1

differs from the result of analytically continuing

f

from

x

to

y

along

C

2

.

This seems to contradict the uniqueness of analytic continuation. This

contradiction

, however, is only apparent because there does not exist a single open set which contains both

C

1

and

C

2

to which

f

can be analytically continued. In fact, in order to accommodate both the analytic continuation along

C

1

and the analytic continuation along

C

2

, one needs to define

f

on a

Riemann surface

. Then there exists an open subset of the Riemann surface which contains the lifts of both

C

1

and

C

2

. Since the lift of the endpoint of the lift of

C

1

lies on a different sheet of the Riemann surface than the endpoint of the lift of

C

2

, there is nothing contradictory about the fact that the analytic continuation of

f

assumes different values at these two distinct points on the Riemann surface.

The notion of analytic continuation along an arc can be extended to analytic continuation along a path. Any path can be written as the union of overlapping arcs. To continue a function along a path, one successively continues it along the various arcs into which the path has been decomposed. It is a simple

consequence

of the rigidity theorem that the result of analytically continuing a function along a path is

independent

of the decomposition of the path into arcs.

Title

analytic continuation

Canonical name

AnalyticContinuation

Date of creation

2014-03-13 16:38:30

Last modified on

2014-03-13 16:38:30

Owner

rspuzio (6075)

Last modified by

pahio (2872)

Numerical id

16

Author

rspuzio (2872)

Entry type

Definition

Classification

msc 30B40

Classification

msc 30A99

Related topic

AnalyticContinuationOfRiemannZeta

Related topic

AnalyticContinuationOfGammaFunction

Defines

analytic continuation along an arc

Defines

analytic continuation along a path

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