natural boundary

natural boundary

It is not always possible to analytically continue a

function

given in a certain region. It might turn
out that, as one approaches the boundary of the region
(or a portion of the boundary), the function always blows
up, so there is no way of extending it past that portion
of the boundary to a larger region. When this happens,
we say that our function has a

natural boundary

.
More formally, we may make a definition as follows:

Definition 1

Let

D

be an open subset of the

complex plane

and let

f

:

D

→

C

be

analytic

.
Then the

natural boundary

of

f

is that subset

B

of

∂

⁡

D

such that, if

z

∈

B

,
then there exists no open neighborhood

N

of

z

and no

analytic function

g

:

N

→

C

such that

f

⁢

(

w

)

=

g

⁢

(

w

)

for all

w

∈

D

∪

N

.

As an example of this phenomenon, consider the

power series

∑

k

=

0

∞

z

k

!

.

By comparison with the

geometric series

, it is seen that this
series

converges absolutely

when

|

z

|

<

1

:

∑

k

=

0

∞

|

z

|

k

!

<

∑

k

=

0

∞

|

z

|

k

=

1

1

-

|

z

|

However, when we try to take the limit

|

z

|

→

1

, we find
that the series

diverges

. Namely, let

p

/

q

be a rational
number and let

r

be a

positive

real variable. Then, if we
set

z

=

r

⁢

exp

⁡

(

2

⁢

i

⁢

π

⁢

p

/

q

)

, then, when

k

≥

q

, we
have that

q

divides

k

!

, so

z

k

!

=

r

k

!

. However,

lim

r

→

1

⁡

∑

k

=

q

∞

r

k

!

diverges, so our powers series diverges when we try to take
the limit

z

→

exp

⁡

(

2

⁢

i

⁢

π

⁢

p

/

q

)

. Since numbers of the
form

exp

⁡

(

2

⁢

i

⁢

π

⁢

p

/

q

)

are dense amongst

complex numbers

with norm

1

, it follows that the limit

z

→

z

0

diverges
whenver

|

z

0

|

=

1

. Hence, the

unit circle

|

z

|

=

1

forms
a natural boundary for the function defined by our power series.

Natural boundaries are not so familiar to beginners because
the functions which one encounters in the more elementary
part of the subject, such as

algebraic functions

,

exponential
functions

, and functions defined by linear differential
equations, do not have natural boundaries. To be sure, one
could technically call a

singular point

a natural boundary,
but this is usually not done, the term “natural boundary”
being reserved for cases where the set on which the
function misbehaves consists of more than just isolated points,
as in the example above.

However, when one gains some more experience and studies more
advanced material, then natural boundaries arise rather
frequently. For instance,

theta functions

,

elliptic modular
functions

, and functions defined by non-linear differential
equations have natural boundaries. Natural boundaries also
play an important role in applications — for instance, in
statistical mechanics, phase transitions (such as freezing
and boiling) are associated with natural boundaries of the
partition function.

Title

natural boundary

Canonical name

NaturalBoundary

Date of creation

2013-03-22 16:49:34

Last modified on

2013-03-22 16:49:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 30A99

Classification

msc 30B40

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