aarjaneiro/planetmath huggingface dataset:
contour integral

contour integral

Let

f

be a complex-valued function defined on the image of a curve (

http://planetmath.org/Curve

)

α

:

[

a

,

b

]

→

ℂ

, let

P

=

{

a

0

,

…

,

a

n

}

be a

partition

(

http://planetmath.org/Partition3

) of

[

a

,

b

]

. We will restrict our attention to

contours

, i.e. curves for which the

parametric equations

consist of a finite number of

continuously differentiable

arcs.
If the sum

∑

i

=

1

n

f

⁢

(

z

i

)

⁢

(

α

⁢

(

a

i

)

-

α

⁢

(

a

i

-

1

)

)

,

where

z

i

is some point

α

⁢

(

t

i

)

such that

a

i

-

1

⩽

t

i

⩽

a

i

,

converges

as

n

tends to

infinity

and the greatest of the numbers

a

i

-

a

i

-

1

tends to zero, then we define the

contour integral

of

f

along

α

to be the

integral

∫

α

f

⁢

(

z

)

⁢

𝑑

z

:=

∫

a

b

f

⁢

(

α

⁢

(

t

)

)

⁢

𝑑

α

⁢

(

t

)

Notes

(i) If

Im

⁡

(

α

)

is a

segment

of the real axis, then this definition reduces to that of the

Riemann integral

of

f

⁢

(

x

)

between

α

⁢

(

a

)

and

α

⁢

(

b

)

.

(ii) An alternative definition, making use of the

Riemann-Stieltjes integral

, is based on the fact that the definition of this can be extended without any other changes in the wording to cover the cases where

f

and

α

are complex-valued functions.

Now let

α

be any curve

[

a

,

b

]

→

ℝ

2

. Then

α

can be expressed in terms of the components

(

α

1

,

α

2

)

and can be associated with the complex-valued function

z

⁢

(

t

)

=

α

1

⁢

(

t

)

+

i

⁢

α

2

⁢

(

t

)

.

Given any complex-valued function of a complex variable,

f

say, defined on

Im

⁡

(

α

)

we define the

contour integral

of

f

along

α

, denoted by

∫

α

f

⁢

(

z

)

⁢

𝑑

z

by

∫

α

f

⁢

(

z

)

⁢

𝑑

z

=

∫

a

b

f

⁢

(

z

⁢

(

t

)

)

⁢

𝑑

z

⁢

(

t

)

whenever the complex Riemann-Stieltjes integral on the right exists.

(iii) Reversing the direction of the curve changes the sign of the integral.

(iv) The contour integral always exists if

α

is

rectifiable

and

f

is

continuous

.

(v) If

α

is

piecewise smooth

and the contour integral of

f

along

α

exists, then

∫

α

f

⁢

𝑑

z

=

∫

a

b

f

⁢

(

z

⁢

(

t

)

)

⁢

z

′

⁢

(

t

)

⁢

𝑑

t

.

Title

contour integral

Canonical name

ContourIntegral

Date of creation

2013-03-22 12:51:44

Last modified on

2013-03-22 12:51:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

23

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 30A99

Classification

msc 30E20

Synonym

complex integral

Synonym

line integral

Synonym

curve integral

Related topic

CauchyIntegralFormula

Related topic

PathIntegral

Related topic

Integral

Related topic

IntegralTransform

Related topic

RealAndImaginaryPartsOfContourIntegral

Defines

contour

Generated on Fri Feb 9 20:13:19 2018 by

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convergence of complex term series

convergence of complex term series

A series

∑

ν

=

1

∞

c

ν

=

c

1

+

c

2

+

c

3

+

…

(1)

with

complex

terms

c

ν

=

a

ν

+

i

b

ν

(

a

ν

,

b

ν

∈

ℝ

∀

ν

)

is

convergent

iff the

sequence

of its

partial sums

converges

to a

complex number

.

Theorem 1.

The series (1) converges iff the series

∑

ν

=

1

∞

a

ν

and

∑

ν

=

1

∞

b

ν

(2)

formed by

real parts

and the

imaginary parts

of its terms both are convergent.

Proof.

Let

ε

>

0

.  Denote

∑

ν

=

1

n

a

ν

:=

s

n

,

∑

ν

=

1

n

b

ν

:=

t

n

,

∑

ν

=

1

n

c

ν

:=

u

n

.

If the series (2) are convergent with sums

S

and

T

, then there is a number

N

such that

|

s

n

-

S

|

<

ε

2

,

|

t

n

-

T

|

<

ε

2

when

n

≧

N

.

Accordingly,

|

u

n

-

(

S

+

i

⁢

T

)

|

=

(

s

n

-

S

)

2

+

(

t

n

-

T

)

2

≦

|

s

n

-

S

|

+

|

t

n

-

T

|

<

ε

when

n

≧

N

,

i.e. the series (1) converges to

S

+

i

⁢

T

.  If, conversely, (1) converges to a complex number

u

=

s

+

i

t

(

s

,

t

∈

ℝ

)

,

then

|

s

n

-

s

|

≦

|

(

s

n

-

s

)

+

i

⁢

(

t

n

-

t

)

|

=

|

u

n

-

u

|

,

|

t

n

-

t

|

≦

|

(

s

n

-

s

)

+

i

⁢

(

t

n

-

t

)

|

=

|

u

n

-

u

|

,

and consequently,

lim

n

→

∞

⁡

s

n

=

s

and

lim

n

→

∞

⁡

t

n

=

t

, i.e. the series (2) are convergent with sums the

real numbers

s

and

t

.

Theorem 2.

The series (1) converges absolutely iff the series (2) both converge absolutely.

Proof.

The

absolute convergence

of (1) means that the series

∑

ν

=

1

∞

|

c

ν

|

converges.  But since

|

c

ν

|

2

=

|

a

ν

|

2

+

|

b

ν

|

2

,  we have

|

a

ν

|

≦

|

c

ν

|

;

|

b

ν

|

≦

|

c

ν

|

≦

|

a

ν

|

+

|

c

ν

|

.

From these

inequalities

we can infer the assertion of the theorem 2.

References

1

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava. Helsinki (1963).

Title

convergence of complex term series

Canonical name

ConvergenceOfComplexTermSeries

Date of creation

2014-10-31 19:04:59

Last modified on

2014-10-31 19:04:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30A99

Classification

msc 40A05

Related topic

OrderOfFactorsInInfiniteProduct

Generated on Fri Feb 9 20:13:22 2018 by

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derivative and differentiability of complex function

derivative and differentiability of complex function

Let

f

⁢

(

z

)

be given uniquely in a neighborhood of the point

z

in

ℂ

.  If the difference quotient

Δ

⁢

f

Δ

⁢

z

=

f

⁢

(

z

+

Δ

⁢

z

)

-

f

⁢

(

z

)

Δ

⁢

z

tends to a finite limit

A

as

Δ

⁢

z

→

0

,
then

A

is the

derivative

of

f

at the point

z

and is denoted by

f

′

⁢

(

z

)

=

A

=

lim

Δ

⁢

z

→

0

⁡

Δ

⁢

f

Δ

⁢

z

.

(1)

Thus the

difference

λ

=

Δ

⁢

f

Δ

⁢

z

-

A

tends to
zero simultaneously with

Δ

⁢

z

, and

Δ

⁢

f

has the expansion

Δ

⁢

f

=

A

⁢

Δ

⁢

z

+

λ

⁢

Δ

⁢

z

.

If we denote

|

Δ

z

|

=

:

ϱ

,  we have

λ

⁢

Δ

⁢

z

=

λ

⁢

Δ

⁢

z

ϱ

⋅

ϱ

=

⟨

ϱ

⟩

⁢

ϱ

where

⟨

ϱ

⟩

means a

complex number

vanishing when

|

Δ

⁢

z

|

=

ϱ

→

0

.  Consequently, (1) implies

Δ

⁢

f

=

A

⁢

Δ

⁢

z

+

⟨

ϱ

⟩

⁢

ϱ

(2)

in which

A

=

f

′

⁢

(

z

)

and

ϱ

=

|

Δ

⁢

z

|

.  It’s easily seen that
the conditions (1) and (2) are

equivalent

.  The latter expresses the

differentiability

of

f

at

z

.  By it one can sayt that the increment of

f

is “locally proportional” to the increment of

z

.  Cf. the
consideration of

differential

of real functions.

References

1

E. Lindelöf

:

Johdatus funktioteoriaan

(‘

Introduction

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

2

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava, Helsinki (1963).

Title

derivative and differentiability of complex function

Canonical

name

DerivativeAndDifferentiabilityOfComplexFunction

Date of creation

2014-02-23 18:20:58

Last modified on

2014-02-23 18:20:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

3

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Generated on Fri Feb 9 20:13:25 2018 by

LaTeXML
examples of periodic functions

examples of periodic functions

We list

common periodic functions

. In the parentheses, there are given their period with least

modulus

.

•

One-periodic

functions

with a real period:

sine (

2

⁢

π

), cosine (

2

⁢

π

),

tangent

(

π

),

cotangent

(

π

), secant (

2

⁢

π

),

cosecant

(

2

⁢

π

), and functions depending on them – especially the

triangular-wave function

(

2

⁢

π

);  the

mantissa function

x

-

⌊

x

⌋

(1).

•

One-periodic functions with an

imaginary

(

http://planetmath.org/ImaginaryNumber

) period:

exponential function

(

2

⁢

i

⁢

π

),

hyperbolic sine

(

2

⁢

i

⁢

π

),

hyperbolic cosine

(

2

⁢

i

⁢

π

),

hyperbolic tangent

(

i

⁢

π

),

hyperbolic cotangent

(

i

⁢

π

), and functions depending on them.

•

Two-periodic functions:

elliptic functions

.

•

Functions with infinitely (

http://planetmath.org/

Infinite

) many periods:

the Dirichlet’s function

f

:

x

↦

{

1

when

⁢

x

∈

ℚ

0

when

⁢

x

∈

ℝ

∖

ℚ

has any

rational number

as its period;  a

constant function

has any number as its period.

Title

examples of periodic functions

Canonical name

ExamplesOfPeriodicFunctions

Date of creation

2013-03-22 17:57:29

Last modified on

2013-03-22 17:57:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Example

Classification

msc 30A99

Classification

msc 26A09

Synonym

common periodic functions

Related topic

PeriodicityOfExponentialFunction

Related topic

HyperbolicIdentities

Related topic

RationalAndIrrational

Related topic

PeriodicFunctions

Related topic

FloorFunction

Related topic

Floor

Generated on Fri Feb 9 20:13:35 2018 by

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exponential integral

exponential integral

The

antiderivative

of the

function

x

↦

e

-

x

x

is not

expressible in closed form

.  Thus such

integrals

(

http://planetmath.org/ImproperIntegral

) as

∫

x

∞

e

-

t

t

⁢

𝑑

t

and

∫

∞

-

x

e

-

t

t

⁢

𝑑

t

,

define certain non-elementary (

http://planetmath.org/ElementaryFunction

)

transcendental functions

.  They are called

exponential integrals

and denoted usually

E

1

and

Ei

, respectively.  Accordingly,

E

1

⁢

(

x

)

:=

∫

x

∞

e

-

t

t

⁢

𝑑

t

Ei

⁢

x

:=

∫

∞

-

x

e

-

t

t

⁢

𝑑

t

=

-

∫

-

x

∞

e

-

t

t

⁢

𝑑

t

:=

∫

-

∞

x

e

-

u

u

⁢

𝑑

u

.

Then one has the connection

E

1

⁢

(

x

)

=

-

Ei

⁢

(

-

x

)

.

For

positive

values of

x

the

series expansion

Ei

⁢

x

=

γ

+

ln

⁡

x

+

∑

j

=

1

∞

x

j

j

!

⁢

j

,

where

γ

is the

http://planetmath.org/node/1883

Euler–Mascheroni constant, is valid.

Note: Some authors use the convention

Ei

⁢

x

:=

∫

x

∞

e

-

t

t

⁢

𝑑

t

.

0.1

Laplace transform of

1

t

+

a

By the

definition

of

Laplace transform

,

ℒ

⁢

{

1

t

+

a

}

=

∫

0

∞

e

-

s

⁢

t

t

+

a

⁢

𝑑

t

.

The substitution (

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

t

+

a

=

u

gives

ℒ

⁢

{

1

t

+

a

}

=

∫

a

∞

e

a

⁢

s

-

s

⁢

u

u

⁢

𝑑

u

=

e

a

⁢

s

⁢

∫

a

∞

e

-

s

⁢

u

u

⁢

𝑑

u

,

from which the substitution

s

⁢

u

=

t

yields

ℒ

⁢

{

1

t

+

a

}

=

e

a

⁢

s

⁢

∫

a

⁢

s

∞

e

-

t

t

⁢

𝑑

t

,

i.e.

ℒ

⁢

{

1

t

+

a

}

=

e

a

⁢

s

⁢

E

1

⁢

(

a

⁢

s

)

.

(1)

Using the rule (

http://planetmath.org/LaplaceTransformOfDerivative

)

ℒ

⁢

{

f

′

⁢

(

t

)

}

=

s

⁢

F

⁢

(

s

)

-

f

⁢

(

0

)

,  one easily derives from (1) the

ℒ

⁢

{

1

(

t

+

a

)

2

}

=

1

a

-

s

⁢

e

a

⁢

s

⁢

E

1

⁢

(

a

⁢

s

)

.

(2)

Title

exponential integral

Canonical

name

ExponentialIntegral

Date of creation

2013-03-22 18:44:17

Last modified on

2013-03-22 18:44:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Classification

msc 26A36

Synonym

Ei

Related topic

LogarithmicIntegral

Related topic

TableOfLaplaceTransforms

Related topic

IndexOfSpecialFunctions

Generated on Fri Feb 9 20:13:38 2018 by

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expressible in closed form

expressible in closed form

An expression is , if it can be converted (simplified) into an expression containing only

elementary functions

, combined by a finite amount of

rational

operations

and

compositions

.
Thus, such a

closed form

must not e.g. limit signs, integral signs, sum signs and “…”.

For example,

∫

d

⁢

x

x

4

+

1

,

may be expressed in the closed form

1

4

⁢

2

⁢

ln

⁡

x

2

+

x

⁢

2

+

1

x

2

-

x

⁢

2

+

1

+

1

2

⁢

2

⁢

arctan

⁡

x

⁢

2

1

-

x

2

+

C

but for

∫

d

⁢

x

x

4

+

1

⁢

𝑑

x

,

there exists no closed form.

In certain

contexts

, the of the “elementary functions” may be enlarged by allowing in it some other

functions

, e.g. the

error function

.

Title

expressible in closed form

Canonical name

ExpressibleInClosedForm

Date of creation

2013-03-22 18:29:09

Last modified on

2013-03-22 18:29:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Classification

msc 26E99

Related topic

ClosedForm

Related topic

IrreducibilityOfBinomialsWithUnityCoefficients

Related topic

ReductionOfEllipticIntegralsToStandardForm

Defines

closed form

Generated on Fri Feb 9 20:13:40 2018 by

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Green functions and conformal mapping

Green functions and conformal mapping

1

Introduction

The

Green function

for the Laplacian operator in two

dimensions

is closely related to

conformal mappings

to the

unit disk

. Given the Green function for a simply-connected region with

Dirichlet boundary conditions

, one can construct the

mapping

by exponentiating the sum of the Green function and its

conjugate harmonic function

. In practise, this can be used to construct

mapping functions

for various regions for which it is possible to solve the

Dirichlet problem

. In principle, it can be used to prove results about conformal mappings using the theory of

differential equations

. For

instance

, one can prove the

Riemann mapping theorem

as a

consequence

of the

existence

of a solution to the Dirichlet problem.

2

Definition

Let

D

be a simply connected subset of the

complex plane

with boundary

∂

⁡

D

and let

a

be a point in the interior of

D

. The Green’s

function

is a function

g

:

D

→

𝐑

such that

1.

g

=

0

on

∂

⁡

D

.

2.

∇

2

⁡

g

=

0

on the interior of

D

.

3.

g

⁢

(

z

)

-

log

⁡

|

z

-

a

|

is

bounded

as

z

approaches

a

.

Note: The third condition actually is

equivalent

to the stronger condition that

g

⁢

(

z

)

-

log

⁡

|

z

-

a

|

is

analytic

at

a

. This follows from the general fact about

harmonic functions

.

3

Construction of the mapping function

Let

h

be the conjugate harmonic function of to

g

. It can be shown that

h

⁢

(

z

)

-

arg

⁡

(

z

-

a

)

is bounded as

z

→

a

and, consequently, that

h

is a multiple-valued function with

branch point

at

a

which increases by

2

⁢

π

every time one encircles

a

.

Now consider the function

f

defined as

e

-

(

g

+

i

⁢

h

)

. This function is

single valued

because, when one circles about

a

, the

argument

of the exponential increases by

2

⁢

π

⁢

i

, but adding

2

⁢

π

⁢

i

to an exponential does not change its value. Since

h

is the conjugate harmonic function of

g

, it follows that

g

+

i

⁢

h

is

holomorphic

and, hence

f

is also holomorphic.

Therefore,

f

maps

D

∖

{

a

}

to

𝐂

. Various things can be said about this mapping.

Because of the maximum princliple,

g

⁢

(

z

)

>

0

for all

z

in the interior of

D

. Hence,

f

maps the interior of

D

into the interior of the unit disk and maps

∂

⁡

D

to the unit circle.

Furthermore, it can be shown that the function

f

is

invertible

so, in fact, it is a

conformal

diffeomorphism

between

D

and the unit disk.

Title

Green functions and conformal mapping

Canonical name

GreenFunctionsAndConformalMapping

Date of creation

2013-03-22 15:57:06

Last modified on

2013-03-22 15:57:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 30A99

Generated on Fri Feb 9 20:13:42 2018 by

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hyperbolic sine integral

hyperbolic sine integral

The

function

hyperbolic sine integral

(in Latin

sinus hyperbolicus integralis

) from

ℝ

to

ℝ

is defined as

Shi

⁡

x

:=

∫

0

x

sinh

⁡

t

t

⁢

𝑑

t

,

(1)

or alternatively as

Shi

⁡

x

:=

∫

0

1

sinh

⁡

t

⁢

x

t

⁢

𝑑

t

.

It isn’t an

elementary function

.  The equation (1) implies the

Taylor series

expansion

Shi

⁡

z

=

z

+

z

3

3

⋅

3

!

+

z

5

5

⋅

5

!

+

z

7

7

⋅

7

!

+

⋯

,

which converges for all

complex

values

z

and thus defines an

entire transcendental function

.
Using the

Taylor expansions

, it is easily seen that

Shi

⁡

x

=

i

⁢

Si

⁡

i

⁢

x

connects Shi to the

sine integral

function.

Shi

⁡

x

satisfies the linear third

differential equation

x

⁢

f

′′′

⁢

(

x

)

+

2

⁢

f

′′

⁢

(

x

)

-

x

⁢

f

′

⁢

(

x

)

=

0

.

Title

hyperbolic sine integral

Canonical name

HyperbolicSineIntegral

Date of creation

2013-03-22 18:27:48

Last modified on

2013-03-22 18:27:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Synonym

Shi

Related topic

HyperbolicFunctions

Related topic

SineIntegral

Generated on Fri Feb 9 20:13:44 2018 by

LaTeXML
identity theorem of holomorphic functions

identity theorem of holomorphic functions

If the

functions

f

and

g

are

holomorphic

in a domain

D

of
the

complex plane

and the equation

f

⁢

(

z

)

=

g

⁢

(

z

)

(1)

is true in an

infinite subset

S

of

D

having an

accumulation point

z

0

in

D

, then (1) is true in the whole

D

.

Remark.

The subset

S

may be e.g. some

neighbourhood

of

z

0

or some arc containing

z

0

.

Title

identity theorem of holomorphic functions

Canonical name

IdentityTheoremOfHolomorphicFunctions

Date of creation

2013-03-22 16:47:05

Last modified on

2013-03-22 16:47:05

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30A99

Synonym

rigidity theorem for analytic functions

Related topic

IdentityTheoremOfPowerSeries

Related topic

IdentityTheorem

Generated on Fri Feb 9 20:13:46 2018 by

LaTeXML
limit for exp(z)

limit for exp(z)

For any

complex number

z

, we have

lim

n

→

∞

⁡

(

1

+

z

n

+

o

⁢

(

1

n

)

)

n

=

exp

⁡

z

,

where

exp

denotes the

exponential function

.

Proof:

For

α

→

0

, we have

ln

⁡

(

1

+

α

)

=

∑

k

=

1

∞

(

-

1

)

k

-

1

⋅

α

k

k

=

α

+

O

⁢

(

α

2

)

.

Therefore

(

1

+

z

n

+

o

⁢

(

1

n

)

)

n

=

exp

⁡

(

n

⁢

ln

⁡

(

1

+

z

n

+

o

⁢

(

1

n

)

)

)

=

exp

⁡

(

n

⁢

(

z

n

+

o

⁢

(

1

n

)

+

O

⁢

(

1

n

2

)

)

)

=

exp

(

z

+

o

(

1

)

+

O

(

1

n

)

)

→

exp

z

for

n

→

∞

.

□

Title

limit for exp(z)

Canonical name

LimitForExpz

Date of creation

2013-03-22 14:34:54

Last modified on

2013-03-22 14:34:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 30A99

Generated on Fri Feb 9 20:13:48 2018 by

LaTeXML
locally bounded

locally bounded

Suppose that

X

is a

topological space

and

Y

a

metric space

.

Definition

.

A set

ℱ

of

functions

f

:

X

→

Y

is said to be

locally bounded

if for
every

x

∈

X

, there exists a

neighbourhood

N

of

x

such that

ℱ

is uniformly bounded on

N

.

In the special case of functions on the

complex plane

where it
is often used, the definition can be given as follows.

Definition

.

A set

ℱ

of functions

f

:

G

⊂

ℂ

→

ℂ

is said to be

locally bounded

if for every

a

∈

G

there exist

constants

δ

>

0

and

M

>

0

such that for all

z

∈

G

such that

|

z

-

a

|

<

δ

,

|

f

⁢

(

z

)

|

<

M

for all

f

∈

ℱ

.

As an example we can look at the set

ℱ

of

entire functions

where

f

⁢

(

z

)

=

z

2

+

t

for any

t

∈

[

0

,

1

]

. Obviously each such

f

is

unbounded

itself, however if we take a small neighbourhood around any point we can
bound all

f

∈

ℱ

. Say on an open ball

B

⁢

(

z

0

,

1

)

we can show
by

triangle inequality

that

|

f

⁢

(

z

)

|

≤

(

|

z

0

|

+

1

)

2

+

1

for all

z

∈

B

⁢

(

z

0

,

1

)

. So this set of functions is locally bounded.

Another example would be say the set of all

analytic functions

from
some region

G

to the

unit disc

. All those functions are

bounded

by 1,
and so we have a uniform bound even over all of

G

.

As a

counterexample

suppose the we take the

constant functions

f

n

⁢

(

z

)

=

n

for
all

natural numbers

n

. While each of these functions is itself bounded,
we can never find a uniform bound for all such functions.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

locally bounded

Canonical name

LocallyBounded

Date of creation

2013-03-22 14:17:47

Last modified on

2013-03-22 14:17:47

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Definition

Classification

msc 30A99

Classification

msc 54-00

Generated on Fri Feb 9 20:13:50 2018 by

LaTeXML
monodromy group

monodromy group

Consider an ordinary

linear differential equation

∑

k

=

0

n

c

k

⁢

(

x

)

⁢

d

n

⁢

y

d

⁢

x

n

=

0

in which the coefficients are polynomials. If

c

n

is not constant, then it is possible that the solutions
of this equation will have

branch points

at the zeros
of

c

n

. (To see if this actually happens, we need to
examine the

indicial equation

.)

By the

persistence of differential equations

, the

analytic
continuation

of a solution of this equation will be another
solution. Pick a

neighborhood

which does not contain any
zeros of

c

n

. Since the

differential equation

is of order

n

, there will be

n

independent

solutions

y

=

f

1

⁢

(

x

)

,

…

,

y

=

f

n

⁢

(

x

)

. (For example, one may exhibit these
solutions as

power series

about some point in the neighborhood.)

Upon analytic continuation back to the original neighborhood via
a chain of neghborhoods, suppose
that the solution

y

=

f

i

⁢

(

x

)

is taken to a solution

y

=

g

i

⁢

(

x

)

.
Because the solutions are

linearly independent

, there will exist a matrix

{

m

i

⁢

j

}

i

,

j

=

1

n

such that

g

i

=

∑

j

=

1

n

m

i

⁢

j

⁢

f

j

.

Now consider the totality of all such matrices corresponding to
all possible ways of making analytic continuations along chains
which begin and end wit the original neighborhood. They form a
group known as the

monodromy group

of the differential
equation. The reason this set is a group is some basic facts
about analytic continuation. First, there is the trivial analytic
continuation which simply takes a

function element

to itself. This
will correspond to the

identity matrix

. Second, we can reverse a
process of analytic continuation. This will correspond to the inverse
matrix. Third, we can follow continuation along one chain of
neighborhoods by continuation along another chain. This will
correspond to multiplying the matrices corresponding to the two chains.

Title

monodromy group

Canonical name

MonodromyGroup

Date of creation

2013-03-22 16:21:51

Last modified on

2013-03-22 16:21:51

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 30A99

Related topic

Monodromy

Generated on Fri Feb 9 20:13:52 2018 by

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