aarjaneiro/planetmath huggingface dataset:
monotonicity of the sequence (1+x/n)^n

monotonicity of the sequence

(

1

+

x

/

n

)

n

Theorem 1

.

Let

x

be a

real number

and let

n

be an integer such that

n

>

0

and

n

+

x

>

0

. Then

(

n

+

x

n

)

n

<

(

n

+

1

+

x

n

+

1

)

n

+

1

.

Proof.

We begin by dividing the two expressions to be compared:

(

n

+

x

+

1

n

+

1

)

n

+

1

(

n

+

x

n

)

n

=

n

+

x

+

1

n

+

1

⋅

(

n

⁢

(

n

+

x

+

1

)

(

n

+

x

)

⁢

(

n

+

1

)

)

n

=

n

+

x

+

1

n

+

1

⋅

(

n

2

+

n

⁢

x

+

n

n

2

+

n

⁢

x

+

n

+

x

)

n

=

n

+

x

+

1

n

+

1

⋅

(

1

-

x

n

2

+

n

⁢

x

+

n

+

x

)

n

Now, when

x

>

0

, we have

0

<

x

n

2

+

n

⁢

x

+

n

+

x

<

1

whilst, when

x

<

0

and

n

+

x

>

0

, we have,

x

n

2

+

n

⁢

x

+

n

+

x

<

0

.

Therefore, we may apply an

inequality for differences of powers

to conclude

(

1

-

x

n

2

+

n

⁢

x

+

n

+

x

)

n

>

1

-

n

⁢

x

n

2

+

n

⁢

x

+

n

+

x

=

n

2

+

n

+

x

n

2

+

n

⁢

x

+

n

+

x

Hence, we have

(

n

+

x

+

1

n

+

1

)

n

+

1

(

n

+

x

n

)

n

>

(

n

+

x

+

1

)

⁢

(

n

2

+

n

+

x

)

(

n

+

1

)

⁢

(

n

2

+

n

⁢

x

+

n

+

x

)

=

n

3

+

2

⁢

n

2

+

n

+

n

2

⁢

x

+

2

⁢

n

⁢

x

+

x

+

x

2

n

3

+

2

⁢

n

2

+

n

+

n

2

⁢

x

+

2

⁢

n

⁢

x

+

x

Note that the numerator is

greater

than the denominator because
it contains every term contained in the denominator and an extra
term

x

2

. Hence this ratio is larger than

1

; multiplying out,
we obtain the

inequality

which was to be demonstrated.
∎

Title

monotonicity of the sequence

(

1

+

x

/

n

)

n

Canonical name

MonotonicityOfTheSequence1Xnn

Date of creation

2013-03-22 17:01:47

Last modified on

2013-03-22 17:01:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

18

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 32A05

Generated on Fri Feb 9 20:31:37 2018 by

LaTeXML
periodicity of exponential function

periodicity of exponential function

Theorem

.

The only periods of the

complex exponential function

z

↦

e

z

are the multiples of

2

⁢

π

⁢

i

.  Thus the

function

is

one-periodic

.

Proof.

Let

ω

be any period of the

exponential function

, i.e.

e

z

+

ω

=

e

z

⁢

e

ω

=

e

z

for all

z

∈

ℂ

.  Because

e

z

is always

≠

0

, we have

e

ω

=

1

.

(1)

If we set

ω

=

:

a

+

i

b

with

a

and

b

reals, (1) gets the form

e

a

⁢

cos

⁡

b

+

i

⁢

e

a

⁢

sin

⁡

b

=

1

,

(2)

which implies (see

equality of complex numbers

)

e

a

⁢

cos

⁡

b

=

1

,

e

a

⁢

sin

⁡

b

=

0

.

As these equations are

squared

and added, we obtain

e

2

⁢

a

=

1

which , since

a

is real, that

a

=

0

.  Thus the preceding equations get the form

cos

⁡

b

=

1

,

sin

⁡

b

=

0

.

These result that

b

=

n

⋅

2

⁢

π

and therefore

ω

=

n

⋅

2

π

i

(

n

=

0

,

±

1

,

±

2

,

±

3

,

…

)

Q.E.D.

References

1

Ernst Lindelöf

:

Johdatus funktioteoriaan

(‘Introduction to function theory’).  Mercatorin kirjapaino, Helsinki (1936).

Title

periodicity

of exponential function

Canonical name

PeriodicityOfExponentialFunction

Date of creation

2014-02-20 14:29:59

Last modified on

2014-02-20 14:29:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Theorem

Classification

msc 32A05

Classification

msc 30D20

Synonym

period of exponential function

Related topic

PeriodicFunctions

Related topic

AnalyticContinuationOfRiemannZetaUsingIntegral

Related topic

ExamplesOfPeriodicFunctions

Related topic

ExponentialFunctionNeverVanishes

Defines

one-periodic

Generated on Fri Feb 9 20:31:39 2018 by

LaTeXML
proof of growth of exponential function

proof of growth of exponential function

In this proof, we first restrict to when

x

and

a

are integers
and only later lift this restricton.

Let

a

>

0

be an integer, let

b

>

1

be real, and let

x

be an
integer.

Consider the following inequality

(

1

+

1

x

)

a

≤

1

+

a

x

⁢

(

1

+

1

x

)

a

-

1

If

x

≥

2

, then we have

(

1

+

1

x

)

a

≤

1

+

a

x

⁢

(

3

2

)

a

-

1

.

Define

X

to be the greater of

2

and

⌈

a

⁢

(

3

/

2

)

a

-

1

/

(

1

-

b

)

⌉

; when

x

>

X

, we have

(

1

+

1

x

)

a

≤

b

.

Rewrite

x

a

/

b

x

as follows when

x

>

X

:

x

a

b

x

=

X

a

b

X

⁢

∏

n

=

X

x

(

1

+

1

n

)

a

⁢

1

b

By the inequality established above, each term in the

product

will be
bounded by

1

/

b

, hence

x

a

b

x

≤

X

a

b

X

⁢

1

(

b

)

x

-

X

Since

b

>

1

, it is also the case that

b

>

1

, hence we have
the inequality

(

b

)

n

≥

1

+

n

⁢

(

b

-

1

)

Combining the last two inequalities yields the following:

x

a

b

x

≤

X

a

b

X

≤

1

1

+

(

x

-

X

)

⁢

(

b

-

1

)

From this, it follows that

lim

x

→

∞

⁡

x

a

/

b

x

=

0

when

a

and

x

are integers.

Now we lift the

restriction

that

a

be an integer. Since the power
function is increasing,

x

a

/

b

x

≤

x

⌈

a

⌉

/

b

x

, so we have

lim

x

→

∞

⁡

x

a

/

b

x

=

0

for real values
of

a

as well.

To lift the restriction on

x

, let us write

x

=

x

1

+

x

2

where

x

1

is an integer and

0

≤

x

2

<

1

. Then we have

x

a

b

x

=

x

1

a

b

x

1

⁢

(

x

1

+

x

2

x

1

)

a

⁢

b

-

x

2

If

x

>

2

, then

(

x

1

+

x

2

)

/

x

2

<

1.5

. Since

x

2

≥

0

,

b

-

x

2

≤

1

. Hence, for all real

x

>

2

, we have

x

a

b

x

≤

1.5

a

⁢

x

1

a

b

x

1

From this inequality, it
follows that

lim

x

→

∞

⁡

x

a

/

b

x

=

0

for real values of

x

as well.

Title

proof of growth of exponential function

Canonical name

ProofOfGrowthOfExponentialFunction

Date of creation

2013-03-22 15:48:36

Last modified on

2013-03-22 15:48:36

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 32A05

Generated on Fri Feb 9 20:31:42 2018 by

LaTeXML
Cauchy integral formula in several variables

Cauchy integral formula in several variables

Let

D

=

D

1

×

…

×

D

n

⊂

ℂ

n

be a

polydisc

.

Theorem

.

Let

f

be a function continuous in

D

¯

(the closure of

D

). Then

f

is

holomorphic

(

http://planetmath.org/HolomorphicFunctionsOfSeveralVariables

) in

D

if and only if
for all

z

=

(

z

1

,

…

,

z

n

)

∈

D

we have

f

⁢

(

z

1

,

…

,

z

n

)

=

1

(

2

⁢

π

⁢

i

)

n

⁢

∫

∂

⁡

D

1

⋯

⁢

∫

∂

⁡

D

n

f

⁢

(

ζ

1

,

…

,

ζ

n

)

(

ζ

1

-

z

1

)

⁢

…

⁢

(

ζ

n

-

z

n

)

⁢

𝑑

ζ

1

⁢

…

⁢

𝑑

ζ

n

.

As in the case of one

variable

this

theorem

can be in fact used as a

definition

of holomorphicity. Note that when

n

>

1

then we are no longer
integrating over the boundary of the polydisc but over the
distinguished boundary, that is over

∂

⁡

D

1

×

…

×

∂

⁡

D

n

.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Cauchy integral formula in several variables

Canonical

name

CauchyIntegralFormulaInSeveralVariables

Date of creation

2013-03-22 15:33:46

Last modified on

2013-03-22 15:33:46

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32A07

Classification

msc 32A10

Generated on Fri Feb 9 20:31:43 2018 by

LaTeXML
logarithmically convex set

logarithmically convex set

Suppose

G

⊂

ℂ

n

, then
we define

log

⁡

∥

G

∥

:=

{

(

log

⁡

|

z

1

|

,

…

,

log

⁡

|

z

n

|

)

∈

ℝ

n

∣

(

z

1

,

…

,

z

n

)

∈

G

}

.

Definition

.

We say

G

⊂

ℂ

n

is a

logarithmically convex set

if

log

⁡

∥

G

∥

⊂

ℝ

n

is a convex set.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

logarithmically convex set

Canonical

name

LogarithmicallyConvexSet

Date of creation

2013-03-22 14:29:32

Last modified on

2013-03-22 14:29:32

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A07

Generated on Fri Feb 9 20:31:45 2018 by

LaTeXML
polydisc

polydisc

Definition

.

We denote the set

D

n

⁢

(

z

,

r

)

:=

{

w

∈

ℂ

n

∣

|

z

k

-

w

k

|

<

r

⁢

for all

⁢

k

=

1

,

…

,

n

}

an

open polydisc

. We can also have

polydiscs

of the form

D

1

⁢

(

z

1

,

r

1

)

×

…

×

D

1

⁢

(

z

n

,

r

n

)

.

The set

∂

⁡

D

1

⁢

(

z

1

,

r

1

)

×

…

×

∂

⁡

D

1

⁢

(

z

n

,

r

n

)

is called the

distinguished boundary

of the polydisc.

Be careful not to confuse this with the open ball in

ℂ

n

as that is defined as

B

⁢

(

z

,

r

)

:=

{

w

∈

ℂ

n

∣

|

z

-

w

|

<

r

}

.

When

n

>

1

then open balls and open polydiscs are not

biholomorphically
equivalent

(there is no 1-1 biholomorphic mapping between the two).

It is common to write

D

¯

n

⁢

(

z

,

r

)

for the closure of the polydisc.
Be careful with this notation however as some texts outside of

complex analysis

use

D

⁢

(

x

,

r

)

and
the “disc” to represent a closed ball in two real dimensions.

Also note that when

n

=

2

the

bidisc

is sometimes used.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

polydisc

Canonical name

Polydisc

Date of creation

2013-03-22 14:29:41

Last modified on

2013-03-22 14:29:41

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A07

Classification

msc 32-00

Synonym

open polydisc

Defines

bidisc

Defines

distinguished boundary

Generated on Fri Feb 9 20:31:47 2018 by

LaTeXML
Reinhardt domain

Reinhardt domain

Definition

.

We call an open set

G

⊂

ℂ

n

a

Reinhardt domain

if

(

z

1

,

…

,

z

n

)

∈

G

implies that

(

e

i

⁢

θ

1

⁢

z

1

,

…

,

e

i

⁢

θ

n

⁢

z

n

)

∈

G

for all real

θ

1

,

…

,

θ

n

.

The reason for studying these kinds of domains is that

logarithmically convex

(

http://planetmath.org/LogarithmicallyConvexSet

)
Reinhardt domain are the domains of

convergence

of

power series

in
several complex variables. Note that in one complex variable, a

Reinhardt domain is just a disc.

Note that the

intersection

of

Reinhardt domains is still a

Reinhardt domain, so for every Reinhardt domain, there is a smallest

Reinhardt domain which contains it.

Theorem

.

Suppose that

G

is a Reinhardt domain which contains 0 and
that

G

~

is the smallest

Reinhardt domain such that

G

⊂

G

~

. Then
any

function

holomorphic

on

G

has a holomorphic
to

G

~

.

It actually turns out that a

Reinhardt domain is a domain of convergence.

examples of

Reinhardt domains in

ℂ

n

are polydiscs such as

𝔻

×

⋯

×

𝔻

⏟

n

where

𝔻

⊂

ℂ

is the

unit disc

.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Reinhardt domain

Canonical name

ReinhardtDomain

Date of creation

2013-03-22 14:29:37

Last modified on

2013-03-22 14:29:37

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A07

Generated on Fri Feb 9 20:31:49 2018 by

LaTeXML
tube domain

tube domain

Definition

.

Suppose

S

⊂

ℝ

n

is an open set (usually

connected

),
then we define

T

S

:=

{

z

∈

ℂ

n

∣

Re

⁡

z

∈

S

}

.

We call

T

S

the

tube domain

associated to

S

.

Basically the idea is that these domains give you

complete

freedom in the

imaginary

coordinates. An interesting fact about these domains is that

T

S

is pseudoconvex if and only if

T

S

is convex in the geometric sense,
and this is true if and only if

S

itself is convex.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

tube domain

Canonical

name

TubeDomain

Date of creation

2013-03-22 14:32:44

Last modified on

2013-03-22 14:32:44

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A07

Generated on Fri Feb 9 20:31:51 2018 by

LaTeXML
Hartogs’s theorem on separate analyticity

Hartogs’s theorem on separate analyticity

Theorem

(Hartogs).

Let

G

⊂

C

n

be an open set and write

z

=

(

z

1

,

…

,

z

n

)

. Let

f

:

G

→

C

be a

function

such that for each

k

=

1

,

…

,

n

and fixed

z

1

,

…

,

z

k

-

1

,

z

k

+

1

,

…

,

z

n

the function

w

↦

f

⁢

(

z

1

,

…

,

z

k

-

1

,

w

,

z

k

+

1

,

…

,

z

n

)

is

holomorphic

on the set

{

w

∈

C

∣

z

1

,

…

,

z

k

-

1

,

w

,

z

k

+

1

,

…

,

z

n

∈

G

}

. Then

f

is

continuous

on

G

.

This is a of an analogue of Goursat’s theorem for several complex variables. That is if we just consider that a function is holomorphic in each
variable separately, then it will turn out to be

continuously differentiable

. Thus we can in fact define

holomorphic functions

of several complex variables to be just functions holomorphic in each variable separately.

Note that there is no analogue of this theorem for real variables. If we
assume that a function

f

:

ℝ

n

→

ℝ

is

differentiable

(or even

analytic

) in each variable separately, it is not true that

f

will necessarily be continuous. The standard example in

ℝ

2

is given by

f

⁢

(

x

,

y

)

=

x

⁢

y

x

2

+

y

2

, then this function has well defined

partial derivatives

in

x

and

y

at 0, but it is not continuous at 0 (try approaching 0 along the line

x

=

y

or

x

=

-

y

).

Even if we assume the function to be smooth (

C

∞

), there is no analogue for real variables. Consider

f

⁢

(

x

,

y

)

=

x

⁢

y

⁢

e

-

1

/

(

x

2

+

y

2

)

,

where we define

f

⁢

(

0

,

0

)

=

0

.

This function is smooth, real analytic in each variable
separately, but it fails to be real analytic at the origin.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Hartogs’s theorem on separate analyticity

Canonical name

HartogssTheoremOnSeparateAnalyticity

Date of creation

2013-03-22 14:29:24

Last modified on

2013-03-22 14:29:24

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32A10

Generated on Fri Feb 9 20:31:54 2018 by

LaTeXML
holomorphic functions of several variables

holomorphic functions of several variables

Definition

.

Let

Ω

⊂

ℂ

n

be a domain and let

f

:

Ω

→

ℂ

be a

function

.

f

is called if it is

holomorphic

(

http://planetmath.org/Holomorphic

) in each variable separately as a function of one variable.

That means that the function

z

k

↦

f

⁢

(

z

1

,

…

,

z

k

,

…

,

z

n

)

is holomorphic as a function of one variable. It is not at all obvious that
such a function is even

continuous

and we must apply the Hartogs’s theorem on separate
analyticity which is not a trivial result.

Historically and some authors today still continue to do so, the definition
of being holomorphic in several variables did include the

continuity

or at
least local boundedness requirement.

Of course we can also characterize

holomorphic functions

by their

power series

.

Proposition

.

f

is holomorphic in

Ω

if and only if near each point

ζ

∈

Ω

there is a neighbourhood

U

and a power series in several variables

∑

α

a

α

⁢

(

z

-

ζ

)

α

,

where

α

ranges over all the
multi-indices,

a

α

∈

C

and such that the series converges to

f

⁢

(

z

)

for

z

∈

U

.

Another way to characterize holomorphic functions is by the use of the

Cauchy-Riemann equations

, which can be given in a very form by the

∂

¯

-operator (

http://planetmath.org/BarpartialOperator

).

Proposition

.

f

is holomorphic if and only if

∂

¯

⁢

f

=

0

.

Despite the

similarities

,
one should be careful about carelessly generalizing results about functions
of one variable to functions of several variables as the theory is quite
different. See the topic entry on several complex variables for more
.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

holomorphic functions of several variables

Canonical name

HolomorphicFunctionsOfSeveralVariables

Date of creation

2013-03-22 15:33:42

Last modified on

2013-03-22 15:33:42

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A10

Generated on Fri Feb 9 20:31:55 2018 by

LaTeXML
multifunction

multifunction

It is common practice among complex analysts to speak of

multiple valued functions

in contexts of “functions” such as

z

.

This somewhat informal notion can be made very precise when the “function” has finitely many values (as the

z

does).

Let

X

and

Y

be sets and denote by

Y

s

⁢

y

⁢

m

m

the

m

th

symmetric power

of

Y

.

Definition

.

A function

f

:

X

→

Y

s

⁢

y

⁢

m

m

is called a

multifunction

,
or an

m

-function from

X

to

Y

, where

m

is the multiplicity.

We can think of the value of

f

at any point as a set of

m

(or fewer) elements.
Let

Y

be a

topological space

(resp.

ℂ

)
A multifunction is said to be

continuous

(resp.

holomorphic

) if all the

elementary symmetric polynomials

of
the elements of

f

are continuous (resp. holomorphic). Equivalently,

f

is continuous (resp. holomorphic)
if it is continuous (resp. holomorphic) as functions to

Y

s

⁢

y

⁢

m

m

≅

Y

m

(resp.

ℂ

s

⁢

y

⁢

m

m

≅

ℂ

m

).

With this definition

z

is a holomorphic multifunction (or a 2-function), into

ℂ

s

⁢

y

⁢

m

2

.

Define the

multigraph

of

f

to be the set:

{

(

x

,

y

)

⁢

∣

X

×

Y

∣

⁢

y

∈

f

⁢

(

x

)

}

.

The multigraph of

z

is the corresponding

Riemann surface

imbedded in

ℂ

2

.

In general, with the aid of the

Weierstrass preparation theorem

we can realize any codimension 1

analytic set

in

ℂ

n

as a multigraph over

ℂ

n

-

1

.

The roots of any

Weierstrass polynomial

(or in general of any monic polynomial with holomorphic coefficients) are a holomorphic multifunction.

References

1

Hassler Whitney.
.
Addison-Wesley, Philippines, 1972.

Title

multifunction

Canonical name

Multifunction

Date of creation

2013-03-22 17:42:08

Last modified on

2013-03-22 17:42:08

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A12

Synonym

m-function

Related topic

SymmetricPower

Related topic

WeierstrassPolynomial

Related topic

MultivaluedFunction

Defines

multigraph

Defines

multiple valued function

Generated on Fri Feb 9 20:32:01 2018 by

LaTeXML
symmetric power

symmetric power

Let

X

be a set and let

X

m

:=

X

×

⋯

×

X

⏟

m

-

times

.

Denote an element of

X

m

by

x

=

(

x

1

,

…

,

x

m

)

.

Define an

equivalence relation

by

x

∼

x

′

if and only if there exists a

permutation

σ

of

(

1

,

…

,

m

)

,

such that

x

i

=

x

σ

⁢

i

′

.

Definition

.

The

m

th

symmetric power

of

X

is
the set

X

s

⁢

y

⁢

m

m

:=

X

m

/

∼

.

That is, the set of

equivalence classes

of

X

m

under the

relation

∼

.

Let

π

be the

natural projection

of

X

m

onto

X

s

⁢

y

⁢

m

m

.

Proposition

.

f

:

X

m

→

Y

is a symmetric function if and only if there exists a

function

g

:

X

s

⁢

y

⁢

m

m

→

Y

such that

f

=

g

∘

π

.

From now on let

R

be an integral domain. Let

τ

′

:

X

m

→

X

m

be the map

τ

′

⁢

(

x

)

:=

(

τ

1

⁢

(

x

)

,

…

,

τ

m

⁢

(

x

)

)

,

where

τ

k

is the

k

th

elementary symmetric
polynomial

. By the above lemma, we have a function

τ

:

X

s

⁢

y

⁢

m

m

→

X

m

, where

τ

′

=

τ

∘

π

.

Proposition

.

τ

is one to one. If

R

is

algebraically closed

, then

τ

is onto.

A very useful case is when

R

=

ℂ

.

In this case, when we put on the natural

complex manifold

structure

onto

ℂ

s

⁢

y

⁢

m

m

,

the map

τ

is a biholomorphism of

ℂ

s

⁢

y

⁢

m

m

and

ℂ

m

.

References

1

Hassler Whitney.
.
Addison-Wesley, Philippines, 1972.

Title

symmetric power

Canonical name

SymmetricPower

Date of creation

2013-03-22 17:42:05

Last modified on

2013-03-22 17:42:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A12

Classification

msc 05E05

Related topic

Multifunction

Generated on Fri Feb 9 20:32:03 2018 by

LaTeXML
Weierstrass polynomial

Weierstrass polynomial

Definition

.

A

function

W

:

ℂ

n

→

ℂ

of the form

W

⁢

(

z

1

,

…

,

z

n

)

=

z

n

m

+

∑

j

=

1

m

-

1

a

j

⁢

(

z

1

,

…

,

z

n

-

1

)

⁢

z

n

j

,

where the

a

j

are

holomorphic functions

in a neighbourhood of the origin, which vanish at the origin,
is called a

Weierstrass polynomial

.

Any codimension 1

complex analytic subvariety

of

ℂ

n

can be written as the zero set
of a Weierstrass polynomial using the

Weierstrass preparation theorem

. This in general cannot be done for higher codimension.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Weierstrass polynomial

Canonical

name

WeierstrassPolynomial

Date of creation

2013-03-22 15:04:25

Last modified on

2013-03-22 15:04:25

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A17

Classification

msc 32B05

Synonym

W-polynomial

Related topic

Multifunction

Related topic

WeierstrassPreparationTheorem

Generated on Fri Feb 9 20:32:05 2018 by

LaTeXML
meromorphic functions of several variables

meromorphic functions of several variables

Definition

.

Let

Ω

⊂

ℂ

n

be a domain and let

h

:

Ω

→

ℂ

be a function.

h

is called if for each

p

∈

Ω

there exists a

neighbourhood

U

⊂

Ω

(

p

∈

U

) and two

holomorphic

(

http://planetmath.org/HolomorphicFunctionsOfSeveralVariables

)
functions

f

,

g

defined in

U

where

g

is not identically zero, such that

h

=

f

/

g

outside the set where

g

=

0

.

Note that

h

is really defined only outside of a

complex analytic subvariety

. Unlike in one

variable

, we cannot simply define

h

to be equal to

∞

at the poles and expect

h

to be a

continuous mapping

to some larger space (the

Riemann sphere

in the case of one variable). The simplest

counterexample

in

ℂ

2

is

(

z

,

w

)

↦

z

/

w

, which does not have a unique limit at the origin. The set of points where there is no unique limit, is called the

indeterminancy set

. That is, the set of points where if

h

=

f

/

g

, and

f

and

g

have no common factors, then the indeterminancy set of

h

is the set where

f

=

g

=

0

.

References

1

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

meromorphic functions of several variables

Canonical

name

MeromorphicFunctionsOfSeveralVariables

Date of creation

2013-03-22 16:01:10

Last modified on

2013-03-22 16:01:10

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Definition

Classification

msc 32A20

Defines

indeterminancy set

Generated on Fri Feb 9 20:32:07 2018 by

LaTeXML