aarjaneiro/planetmath huggingface dataset:
holomorphic functions and Laplace’s equation

holomorphic functions and Laplace’s equation

It can be easily shown that the real and

imaginary parts

of any

holomorphic functions

(that is,

functions

satisfying the

Cauchy-Riemann equations

) separately satisfy Laplace’s equation. Consider the Cauchy-Riemann equations:

∂

⁡

u

∂

⁡

x

=

∂

⁡

v

∂

⁡

y

∂

⁡

u

∂

⁡

y

=

-

∂

⁡

v

∂

⁡

x

Now

differentiate

the first equation with respect to

x

and the second with respect to

y

:

∂

2

⁡

u

∂

⁡

x

2

=

∂

2

⁡

v

∂

⁡

x

⁢

∂

⁡

y

∂

2

⁡

u

∂

⁡

y

2

=

-

∂

2

⁡

v

∂

⁡

y

⁢

∂

⁡

x

Now add both equations together:

∂

2

⁡

u

∂

⁡

x

2

+

∂

2

⁡

u

∂

⁡

y

2

=

∂

2

⁡

v

∂

⁡

x

⁢

∂

⁡

y

-

∂

2

⁡

v

∂

⁡

y

⁢

∂

⁡

x

v

must be

continuous

, as it is holomorphic, and the mixed

derivatives

of

continuous functions

are equal. Hence:

∂

2

⁡

u

∂

⁡

x

2

+

∂

2

⁡

u

∂

⁡

y

2

=

0

∇

2

⁡

u

=

0

The same process, repeated for

v

, yields Laplace’s equation for

v

.

u

and

v

are

harmonic functions

, as they satisfy Laplace’s equation, and they are referred to as

conjugate harmonics

. Functions satisfying Laplace’s equation are important in electromagnetism, and the search for harmonic functions forms part of

potential theory

.

Title

holomorphic functions and Laplace’s equation

Canonical name

HolomorphicFunctionsAndLaplacesEquation

Date of creation

2013-03-22 17:47:32

Last modified on

2013-03-22 17:47:32

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

7

Author

invisiblerhino (19637)

Entry type

Application

Classification

msc 30E99

Related topic

PotentialTheory

Generated on Fri Feb 9 20:22:03 2018 by

LaTeXML
identity theorem

identity theorem

Identity

theoremFernando Sanz Gamiz

Lemma 1

.

Let

f

be

analytic

on

Ω

⊆

C

and let

L

be the set of

accumulation points

(also called

limit points

or

cluster points

) of

{

z

∈

Ω

:

f

⁢

(

z

)

=

0

}

in

Ω

. Then

L

is both open
and closed in

Ω

.

Proof.

By definition of accumulation point,

L

is closed. To see that it
is also open, let

z

0

∈

L

, choose an

open ball

B

⁢

(

z

0

,

r

)

⊆

Ω

and write

f

⁢

(

z

)

=

∑

n

=

0

∞

a

n

⁢

(

z

-

z

0

)

n

,

z

∈

B

⁢

(

z

0

,

r

)

. Now

f

⁢

(

z

0

)

=

0

, and hence either

f

has a zero
of order

m

at

z

0

(for some

m

), or else

a

n

=

0

for all

n

.
In the former case, there is a

function

g analytic on

Ω

such
that

f

⁢

(

z

)

=

(

z

-

z

0

)

m

⁢

g

⁢

(

z

)

,

z

∈

Ω

, with

g

⁢

(

z

0

)

≠

0

. By

continuity

of

g

,

g

⁢

(

z

)

≠

0

for all

z

sufficiently close to

z

0

, and consequently

z

0

is an

isolated point

of

{

z

∈

Ω

:

f

⁢

(

z

)

=

0

}

. But then

z

0

∉

L

, contradicting our

assumption

. Thus, it must be the case that

a

n

=

0

for all n, so
that

f

≡

0

on

B

⁢

(

z

0

,

r

)

. Consequently,

B

⁢

(

z

0

,

r

)

∈

L

,
proving that

L

is open in

Ω

.
∎

Theorem 1

(Identity theorem).

Let

Ω

be a open

connected

subset of

C

(i.e., a domain). If

f

and

g

are analytic on

Ω

and

{

z

∈

Ω

:

f

⁢

(

z

)

=

g

⁢

(

z

)

}

has an accumulation point in

Ω

, then

f

≡

g

on

Ω

.

Proof.

We have that

{

z

∈

Ω

:

f

⁢

(

z

)

-

g

⁢

(

z

)

=

0

}

has an
accumulation point, hence, according to the previous lemma, it is
open and closed (also called ”clopen”). But, as

Ω

is
connected, the only

closed and open

subset at once is

Ω

itself, therefore

{

z

∈

Ω

:

f

⁢

(

z

)

-

g

⁢

(

z

)

=

0

}

=

Ω

,
i.e.,

f

≡

g

on

Ω

.
∎

Remark 1

.

This

theorem

provides a very powerful and useful tool to test
whether two

analytic functions

, whose values coincide in some
points, are indeed the same function. Namely, unless the points in
which they are equal are isolated, they are the same function.

Title

identity theorem

Canonical name

IdentityTheorem

Date of creation

2013-03-22 17:10:38

Last modified on

2013-03-22 17:10:38

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

8

Author

fernsanz (8869)

Entry type

Theorem

Classification

msc 30E99

Related topic

Complex

Related topic

ZeroesOfAnalyticFunctionsAreIsolated

Related topic

TopologyOfTheComplexPlane

Related topic

ClopenSubset

Related topic

IdentityTheoremOfHolomorphicFunctions

Related topic

PlacesOfHolomorphicFunction

Generated on Fri Feb 9 20:22:05 2018 by

LaTeXML
Paley-Wiener theorem

Paley-Wiener theorem

Let

f

⁢

(

z

)

be an

entire function

such that

|

f

⁢

(

z

)

|

≤

K

⁢

e

γ

⁢

|

z

|

for some

K

≥

0

and

γ

>

0

. If the

restriction

of

f

to the real line
is in

L

2

⁢

(

ℝ

)

, then there exists a

function

g

⁢

(

t

)

∈

L

2

⁢

(

-

γ

,

γ

)

such that

f

⁢

(

z

)

=

1

2

⁢

π

⁢

∫

-

γ

γ

g

⁢

(

t

)

⁢

e

i

⁢

z

⁢

t

⁢

𝑑

t

for all

z

.

Title

Paley-Wiener theorem

Canonical name

PaleyWienerTheorem

Date of creation

2013-03-22 15:25:42

Last modified on

2013-03-22 15:25:42

Owner

Gorkem (3644)

Last modified by

Gorkem (3644)

Numerical id

14

Author

Gorkem (3644)

Entry type

Theorem

Classification

msc 30E99

Generated on Fri Feb 9 20:22:07 2018 by

LaTeXML
proof of the Cauchy-Riemann equations

proof of the Cauchy-Riemann equations

Existence of complex derivative implies the Cauchy-Riemann
equations.

Suppose that the

complex
derivative

f

′

⁢

(

z

)

=

lim

ζ

→

0

⁡

f

⁢

(

z

+

ζ

)

-

f

⁢

(

z

)

ζ

(1)

exists for some

z

∈

ℂ

.
This means that for all

ϵ

>

0

, there exists a

ρ

>

0

, such that
for all

complex

ζ

with

|

ζ

|

<

ρ

, we have

|

f

′

⁢

(

z

)

-

f

⁢

(

z

+

ζ

)

-

f

⁢

(

z

)

ζ

|

<

ϵ

.

Henceforth, set

f

=

u

+

i

⁢

v

,

z

=

x

+

i

⁢

y

.

If

ζ

is
real, then the above limit reduces to a

partial derivative

in

x

, i.e.

f

′

⁢

(

z

)

=

∂

⁡

f

∂

⁡

x

=

∂

⁡

u

∂

⁡

x

+

i

⁢

∂

⁡

v

∂

⁡

x

,

Taking the limit with an

imaginary

ζ

we deduce that

f

′

⁢

(

z

)

=

-

i

⁢

∂

⁡

f

∂

⁡

y

=

-

i

⁢

∂

⁡

u

∂

⁡

y

+

∂

⁡

v

∂

⁡

y

.

Therefore

∂

⁡

f

∂

⁡

x

=

-

i

⁢

∂

⁡

f

∂

⁡

y

,

and breaking this

relation

up into its real and

imaginary parts

gives
the

Cauchy-Riemann equations

.

The Cauchy-Riemann
equations imply the existence of a complex derivative.

Suppose that the Cauchy-Riemann
equations

∂

⁡

u

∂

⁡

x

=

∂

⁡

v

∂

⁡

y

,

∂

⁡

u

∂

⁡

y

=

-

∂

⁡

v

∂

⁡

x

,

hold for a fixed

(

x

,

y

)

∈

ℝ

2

,
and that all the
partial derivatives are

continuous at

(

x

,

y

)

as well. The

continuity

implies that all

directional derivatives

exist as well. In
other words, for

ξ

,

η

∈

ℝ

and

ρ

=

ξ

2

+

η

2

we have

u

⁢

(

x

+

ξ

,

y

+

η

)

-

u

⁢

(

x

,

y

)

-

(

ξ

⁢

∂

⁡

u

∂

⁡

x

+

η

⁢

∂

⁡

u

∂

⁡

y

)

ρ

→

0

,

as

⁢

ρ

→

0

,

with a similar relation holding for

v

⁢

(

x

,

y

)

. Combining the two scalar
relations into a vector relation we obtain

ρ

-

1

⁢

∥

(

u

⁢

(

x

+

ξ

,

y

+

η

)

v

⁢

(

x

+

ξ

,

y

+

η

)

)

-

(

u

⁢

(

x

,

y

)

v

⁢

(

x

,

y

)

)

-

(

∂

⁡

u

∂

⁡

x

∂

⁡

u

∂

⁡

y

∂

⁡

v

∂

⁡

x

∂

⁡

v

∂

⁡

y

)

⁢

(

ξ

η

)

∥

→

0

,

as

⁢

ρ

→

0

.

Note that
the Cauchy-Riemann equations imply that the matrix-vector

product

above is equivalent to the product of two

complex numbers

, namely

(

∂

⁡

u

∂

⁡

x

+

i

⁢

∂

⁡

v

∂

⁡

x

)

⁢

(

ξ

+

i

⁢

η

)

.

Setting

f

⁢

(

z

)

=

u

⁢

(

x

,

y

)

+

i

⁢

v

⁢

(

x

,

y

)

,

f

′

⁢

(

z

)

=

∂

⁡

u

∂

⁡

x

+

i

⁢

∂

⁡

v

∂

⁡

x

ζ

=

ξ

+

i

⁢

η

we can therefore rewrite the above limit relation as

|

f

⁢

(

z

+

ζ

)

-

f

⁢

(

z

)

-

f

′

⁢

(

z

)

⁢

ζ

ζ

|

→

0

,

as

⁢

ρ

→

0

,

which is the complex limit definition of

f

′

⁢

(

z

)

shown
in (

1

).

Title

proof of the Cauchy-Riemann equations

Canonical name

ProofOfTheCauchyRiemannEquations

Date of creation

2013-03-22 12:55:39

Last modified on

2013-03-22 12:55:39

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Proof

Classification

msc 30E99

Defines

complex derivative

Generated on Fri Feb 9 20:22:10 2018 by

LaTeXML
removable singularity

removable singularity

Let

U

⊂

ℂ

be an open neighbourhood of a
point

a

∈

ℂ

. We say that a

function

f

:

U

\

{

a

}

→

ℂ

has a

removable singularity

at

a

, if the

complex derivative

f

′

⁢

(

z

)

exists for all

z

≠

a

, and
if

f

⁢

(

z

)

is

bounded

near

a

.

Removable singularities can, as the name suggests, be removed.

Theorem 1

Suppose that

f

:

U

\

{

a

}

→

C

has a removable
singularity at

a

. Then,

f

⁢

(

z

)

can be holomorphically extended to
all of

U

, i.e.
there exists a

holomorphic

g

:

U

→

C

such that

g

⁢

(

z

)

=

f

⁢

(

z

)

for all

z

≠

a

.

Proof.

Let

C

be a circle centered at

a

, oriented counterclockwise, and
sufficiently small so that

C

and its interior are contained in

U

. For

z

in the interior of

C

, set

g

⁢

(

z

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

ζ

-

z

⁢

𝑑

ζ

.

Since

C

is a

compact set

, the defining limit for the

derivative

d

d

⁢

z

⁢

f

⁢

(

ζ

)

ζ

-

z

=

f

⁢

(

ζ

)

(

ζ

-

z

)

2

converges uniformly

for

ζ

∈

C

. Thanks to the

uniform
convergence

, the order of the derivative and the integral operations
can be interchanged. Hence, we may deduce that

g

′

⁢

(

z

)

exists
for all

z

in the interior of

C

. Furthermore, by the

Cauchy
integral formula

we have that

f

⁢

(

z

)

=

g

⁢

(

z

)

for all

z

≠

a

, and therefore

g

⁢

(

z

)

furnishes us with the desired extension.

Title

removable singularity

Canonical name

RemovableSingularity

Date of creation

2013-03-22 12:56:01

Last modified on

2013-03-22 12:56:01

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Definition

Classification

msc 30E99

Related topic

EssentialSingularity

Generated on Fri Feb 9 20:22:19 2018 by

LaTeXML
Riemann’s theorem on isolated singularities

Riemann’s theorem on isolated singularities

Let the

complex function

f

⁢

(

z

)

be

holomorphic

in a deleted

neighbourhood

of the point

z

=

z

0

of the

closed complex plane

ℂ

∪

{

∞

}

.  This point is

•

a

regular point

(or a

removable singularity

) iff

f

⁢

(

z

)

is

bounded

in a neighbourhood of

z

0

,

•

a pole iff

lim

z

→

z

0

⁡

|

f

⁢

(

z

)

|

=

+

∞

,

•

an

essential singularity

iff there is neither of the above cases.

Title

Riemann’s theorem on

isolated singularities

Canonical name

RiemannsTheoremOnIsolatedSingularities

Date of creation

2013-03-22 19:00:51

Last modified on

2013-03-22 19:00:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E99

Synonym

Riemann’s theorem

Related topic

RiemannsRemovableSingularityTheorem

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

LaTeXML
operator

operator

Let

G

⊂

ℂ

n

be a domain and let

f

:

G

→

ℂ

be a

C

1

function

(

continuously differentiable

)

(

z

1

,

…

,

z

n

)

↦

f

⁢

(

z

1

,

…

,

z

n

)

where

z

j

=

x

j

+

i

⁢

y

j

.
We can think of

G

as a subset of

ℝ

2

⁢

n

.
We therefore
have the following

partial derivatives

for all

1

≤

j

≤

n

,

∂

⁡

f

∂

⁡

z

j

:=

1

2

⁢

(

∂

⁡

f

∂

⁡

x

j

-

i

⁢

∂

⁡

f

∂

⁡

y

j

)

,

∂

⁡

f

∂

⁡

z

¯

j

:=

1

2

⁢

(

∂

⁡

f

∂

⁡

x

j

+

i

⁢

∂

⁡

f

∂

⁡

y

j

)

.

Now let

d

be the standard

exterior derivative

on

ℝ

2

⁢

n

and the

d

⁢

x

j

and

d

⁢

y

j

the standard basis of cotangent
vectors. Then if we define

d

⁢

z

j

:=

d

⁢

x

j

+

i

⁢

d

⁢

y

j

,

d

⁢

z

¯

j

:=

d

⁢

x

j

-

i

⁢

d

⁢

y

j

,

then we can define two new operators acting on

C

1

functions on

G

giving 1-forms by

∂

⁡

f

:=

∑

j

=

1

n

∂

⁡

f

∂

⁡

z

j

⁢

d

⁢

z

j

,

∂

¯

⁢

f

:=

∑

j

=

1

n

∂

⁡

f

∂

⁡

z

¯

j

⁢

d

⁢

z

¯

j

.

By direct calculation we immediately see that

d

⁢

f

=

∂

⁡

f

+

∂

¯

⁢

f

.

Similarly we now define

∂

and

∂

¯

on arbitrary

differential form

ω

=

∑

α

,

β

f

α

,

β

⁢

d

⁢

z

α

∧

d

⁢

z

¯

β

, where

α

and

β

range over all multi-indices with
elements less then

n

, where if

α

=

(

α

1

,

…

,

α

k

)

then

d

⁢

z

α

=

d

⁢

z

α

1

∧

…

∧

d

⁢

z

α

k

,
and

f

α

,

β

is a

C

1

,

complex

valued function
on

G

.

∂

⁡

ω

:=

∑

α

,

β

∂

⁡

f

α

,

β

∂

⁡

z

j

⁢

d

⁢

z

j

∧

d

⁢

z

α

∧

d

⁢

z

¯

β

,

∂

¯

⁢

ω

:=

∑

α

,

β

∂

⁡

f

α

,

β

∂

⁡

z

¯

j

⁢

d

⁢

z

¯

j

∧

d

⁢

z

α

∧

d

⁢

z

¯

β

.

Again a direct calculation shows that

d

=

∂

+

∂

¯

.

The

Cauchy-Riemann equations

are then given by

∂

¯

⁢

f

=

0

That is,

f

is

holomorphic

if and only if it satisfies the above equations.
Note that this only applies to functions. If

∂

¯

⁢

ω

=

0

for a differential form, then the

coefficients

in the standard basis
need not be holomorphic.

Proposition

.

∂

¯

and

∂

satisfy the following properties

•

∂

¯

and

∂

are linear,

•

∂

¯

2

=

∂

¯

⁢

∂

¯

=

0

and

∂

2

=

∂

⁡

∂

=

0

,

•

∂

¯

⁢

∂

-

∂

⁡

∂

¯

=

0

.

While

∂

¯

⁢

u

=

0

is our condition for

u

to be a

holomorphic function

it turns out that it is more important to solve the inhomogeneous

∂

¯

⁢

u

=

f

equation, as that allows us to construct holomorphic
objects from nonholomorphic ones.

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

operator

Canonical name

barpartialOperator

Date of creation

2013-03-22 15:10:39

Last modified on

2013-03-22 15:10:39

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 30E99

Classification

msc 32A99

Synonym

d bar operator

Synonym

d-bar operator

Defines

∂

operator

Generated on Fri Feb 9 20:22:23 2018 by

LaTeXML
uniformization theorem

uniformization theorem

Every simply connected

Riemann surface

X

is

biholomorphic

either to

ℙ

1

,

ℂ

or the

unit disk

Δ

.

Title

uniformization theorem

Canonical name

UniformizationTheorem

Date of creation

2013-03-22 15:37:48

Last modified on

2013-03-22 15:37:48

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

5

Author

Simone (5904)

Entry type

Theorem

Classification

msc 30F10

Classification

msc 30F20

Generated on Fri Feb 9 20:22:25 2018 by

LaTeXML
a harmonic function on a graph which is bounded below and nonconstant

a harmonic function on a graph which is bounded below and nonconstant

There exists no

harmonic function

on all of the

d

-dimensional grid

ℤ

d

which is bounded below and nonconstant. This categorises a particular property of the grid; below we see that other graphs can admit such harmonic functions.

Let

𝒯

3

=

(

V

3

,

E

3

)

be a 3-regular tree. Assign “levels” to the vertices of

𝒯

3

as follows: Fix a vertex

o

∈

V

3

, and let

π

be a branch of

𝒯

3

(an infinite

simple path

) from

o

. For every vertex

v

∈

V

3

of

𝒯

3

there exists a

unique

shortest path from

v

to a vertex of

π

; let

ℓ

⁢

(

v

)

=

|

π

|

be the length of this path.

Now define

f

⁢

(

v

)

=

2

-

ℓ

⁢

(

v

)

>

0

. Without loss of generality, note that the three neighbours

u

1

,

u

2

,

u

3

of

v

satisfy

ℓ

⁢

(

u

1

)

=

ℓ

⁢

(

v

)

-

1

(“

u

1

is the parent of

v

”),

ℓ

⁢

(

u

2

)

=

ℓ

⁢

(

u

3

)

=

ℓ

⁢

(

v

)

+

1

(“

u

2

,

u

3

are the

siblings

of

v

”). And indeed,

1

3

⁢

(

2

ℓ

⁢

(

v

)

-

1

+

2

ℓ

⁢

(

v

)

+

1

+

2

ℓ

⁢

(

v

)

+

1

)

=

2

ℓ

⁢

(

v

)

.

So

f

is a positive nonconstant harmonic function on

𝒯

3

.

Title

a harmonic function on a graph which is bounded below and nonconstant

Canonical name

AHarmonicFunctionOnAGraphWhichIsBoundedBelowAndNonconstant

Date of creation

2013-03-22 12:44:26

Last modified on

2013-03-22 12:44:26

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Example

Classification

msc 30F15

Classification

msc 31C05

Classification

msc 31B05

Classification

msc 31A05

Generated on Fri Feb 9 20:22:27 2018 by

LaTeXML
example of harmonic functions on graphs

example of harmonic functions on graphs

1.

Let

G

=

(

V

,

E

)

be a

connected

finite graph

, and let

a

,

z

∈

V

be two of its vertices. The

function

f

⁢

(

v

)

=

ℙ

⁢

{

simple random walk from

v

hits

a

before

z

}

is a

harmonic function

except on

{

a

,

z

}

.

Finiteness of

G

is required only to ensure

f

is

well-defined

. So we may replace “

G

finite” with “

simple random walk

on

G

is recurrent”.

2.

Let

G

=

(

V

,

E

)

be a graph, and let

V

′

⊆

V

. Let

α

:

V

′

→

ℝ

be some

boundary condition

. For

u

∈

V

, define a

random variable

X

u

to be the first vertex of

V

′

that simple random walk from

u

hits. The function

f

⁢

(

v

)

=

𝔼

⁡

α

⁢

(

X

v

)

is a harmonic function except on

V

′

.

The first example is a special case of this one, taking

V

′

=

{

a

,

z

}

and

α

⁢

(

a

)

=

1

,

α

⁢

(

z

)

=

0

.

Title

example of harmonic functions on graphs

Canonical name

ExampleOfHarmonicFunctionsOnGraphs

Date of creation

2013-03-22 12:45:53

Last modified on

2013-03-22 12:45:53

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Example

Classification

msc 30F15

Classification

msc 31C05

Classification

msc 31B05

Classification

msc 31A05

Generated on Fri Feb 9 20:22:29 2018 by

LaTeXML
harmonic conjugate function

harmonic conjugate function

Two

harmonic functions

u

and

v

from an open (

http://planetmath.org/OpenSet

) subset

A

of

ℝ

×

ℝ

to

ℝ

, which satisfy the

Cauchy-Riemann equations

u

x

=

v

y

,

u

y

=

-

v

x

,

(1)

are the

harmonic conjugate functions

of each other.

•

The relationship between

u

and

v

has a geometric meaning:  Let’s determine the slopes of the constant-value curves

u

⁢

(

x

,

y

)

=

a

and

v

⁢

(

x

,

y

)

=

b

in any point

(

x

,

y

)

by differentiating these equations.  The first gives

u

x

⁢

d

⁢

x

+

u

y

⁢

d

⁢

y

=

0

,  or

d

⁢

y

d

⁢

x

(

u

)

=

-

u

x

u

y

=

tan

⁡

α

,

and the second similarly

d

⁢

y

d

⁢

x

(

v

)

=

-

v

x

v

y

but this is, by virtue of (1), equal to

u

y

u

x

=

-

1

tan

⁡

α

.

Thus, by the

condition of orthogonality

, the curves intersect at

right angles

in every point.

•

If one of

u

and

v

is known, then the other may be determined with (1):  When e.g. the

function

u

is known, we need only to the

line integral

v

⁢

(

x

,

y

)

=

∫

(

x

0

,

y

0

)

(

x

,

y

)

(

-

u

y

⁢

d

⁢

x

+

u

x

⁢

d

⁢

y

)

along any connecting

(

x

0

,

y

0

)

and

(

x

,

y

)

in

A

.  The result is the

harmonic conjugate

v

of

u

, unique up to a real addend if

A

is simply connected.

•

It follows from the preceding, that every harmonic function has a harmonic conjugate function.

•

The

real part

and the imaginary part of a

holomorphic function

are always the harmonic conjugate functions of each other.

Example.

sin

⁡

x

⁢

cosh

⁡

y

and

cos

⁡

x

⁢

sinh

⁡

y

are harmonic conjugates of each other.

Title

harmonic conjugate function

Canonical name

HarmonicConjugateFunction

Date of creation

2013-03-22 14:45:11

Last modified on

2013-03-22 14:45:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

21

Author

pahio (2872)

Entry type

Definition

Classification

msc 30F15

Classification

msc 31A05

Synonym

harmonic conjugate

Synonym

conjugate harmonic function

Synonym

conjugate harmonic

Related topic

ComplexConjugate

Related topic

OrthogonalCurves

Related topic

TopicEntryOnComplexAnalysis

Related topic

ExactDifferentialEquation

Generated on Fri Feb 9 20:22:31 2018 by

LaTeXML
Harnack’s principle

Harnack’s principle

If the

functions

u

1

⁢

(

z

)

,

u

2

⁢

(

z

)

, …  are harmonic (

http://planetmath.org/HarmonicFunction

) in the domain

G

⊆

ℂ

and

u

1

⁢

(

z

)

≤

u

2

⁢

(

z

)

≤

⋯

in every point of

G

, then

lim

n

→

∞

⁡

u

n

⁢

(

z

)

either is

infinite

in every point of the domain or it is finite in every point of the domain, in both cases uniformly (

http://planetmath.org/UniformConvergence

) in each closed (

http://planetmath.org/ClosedSet

) subdomain of

G

.  In the latter case, the function

u

⁢

(

z

)

=

lim

n

→

∞

⁡

u

n

⁢

(

z

)

is harmonic in the domain

G

(cf. limit function of

sequence

).

Title

Harnack’s principle

Canonical name

HarnacksPrinciple

Date of creation

2013-03-22 14:57:35

Last modified on

2013-03-22 14:57:35

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

14

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 30F15

Classification

msc 31A05

Generated on Fri Feb 9 20:22:32 2018 by

LaTeXML