aarjaneiro/planetmath huggingface dataset:
Hermite equation

Hermite equation

The

linear differential equation

d

2

⁢

f

d

⁢

z

2

-

2

⁢

z

⁢

d

⁢

f

d

⁢

z

+

2

⁢

n

⁢

f

=

0

,

in which

n

is a real , is called the

Hermite equation

.  Its

general solution

is

f

:=

A

⁢

f

1

+

B

⁢

f

2

with

A

and

B

arbitrary and the

functions

f

1

and

f

2

presented as

f

1

⁢

(

z

)

:=

z

+

2

⁢

(

1

-

n

)

3

!

⁢

z

3

+

2

2

⁢

(

1

-

n

)

⁢

(

3

-

n

)

5

!

⁢

z

5

+

2

3

⁢

(

1

-

n

)

⁢

(

3

-

n

)

⁢

(

5

-

n

)

7

!

⁢

z

7

+

…

,

f

2

⁢

(

z

)

:=

1

+

2

⁢

(

-

n

)

2

!

⁢

z

2

+

2

2

⁢

(

-

n

)

⁢

(

2

-

n

)

4

!

⁢

z

4

+

2

3

⁢

(

-

n

)

⁢

(

2

-

n

)

⁢

(

4

-

n

)

6

!

⁢

z

6

+

…

It’s easy to check that these

power series

satisfy the

differential equation

.  The

coefficients

b

ν

in both series obey the recurrence

b

ν

=

2

⁢

(

ν

-

2

-

n

)

ν

⁢

(

n

⁢

u

-

1

)

⁢

b

ν

-

2

.

Thus we have the radii of

convergence

(

http://planetmath.org/RadiusOfConvergence

)

R

=

lim

ν

→

∞

⁡

|

b

ν

-

2

b

ν

|

=

lim

ν

→

∞

⁡

ν

2

⋅

1

-

1

/

ν

1

-

(

n

+

2

)

/

ν

=

∞

.

Therefore the series

converge

in the whole

complex plane

and define

entire functions

.

If the

n

is a non-negative integer, then one of

f

1

and

f

2

is simply a polynomial function.  The

polynomial

solutions of the Hermite equation are usually normed so that the highest degree (

http://planetmath.org/PolynomialRing

) is

(

2

⁢

z

)

n

and called the

Hermite polynomials

.

Title

Hermite equation

Canonical name

HermiteEquation

Date of creation

2013-03-22 15:16:15

Last modified on

2013-03-22 15:16:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Definition

Classification

msc 34M05

Synonym

Hermite differential equation

Related topic

ChebyshevEquation

Generated on Fri Feb 9 20:42:51 2018 by

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inverse Galois problem

inverse Galois problem

The

inverse Galois problem

is extremely to , yet one of the hardest problems for theorists and . It generally asks for descriptions of the of groups that can occur as

Galois groups

. Of course, a significantly more precise formulation is required, for example, because of a result that that every Galois group is

profinite

, and every profinite group is a Galois group. Also ambiguous is what field(s) we allow ourselves to include when computing the Galois group. Unfortunately, many of these related questions all go under the heading “the inverse Galois problem,” so care must be taken to determine an exact formulation of the question being asked.

As an example of a partial

solution

to this question, it is known that every finite abelian group occurs as the Galois group of an

extension

over

ℚ

(by the

Kronecker-Weber theorem

), though it is

not

known whether or not this is true for every finite (not necessarily

abelian

) group. This latter question can also be phrased in of the

absolute Galois group

: “Does every

finite group

occur as a

quotient group

of the absolute Galois group

Gal

⁢

(

ℚ

¯

/

ℚ

)

”? Thus, an answer to this question would not only reveal information about the nature of finite Galois groups, but also shed light on one of the most elusive objects in all of

algebra

and

number theory

.

It is also known (see Shafarevich’ theorem) that every

solvable group

occurs as a Galois group.

Title

inverse Galois problem

Canonical name

InverseGaloisProblem

Date of creation

2013-03-22 15:01:14

Last modified on

2013-03-22 15:01:14

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 34M50

Classification

msc 13B05

Classification

msc 11R32

Related topic

ShafarevichsTheorem

Generated on Fri Feb 9 20:42:53 2018 by

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connection between Riccati equation and Airy functions

connection between Riccati equation and Airy functions

We report an interesting connection relating

Riccati equation

with Airy functions. Let us consider the nonlinear complex operator

𝔏

:

z

∈

ℂ

↦

ζ

with kernel given by

d

⁢

ζ

d

⁢

z

+

ζ

2

+

a

⁢

(

z

)

⁢

ζ

+

b

⁢

(

z

)

=

0

,

(1)

a nonlinear ODE of the first order so-called Riccati equation. In order to accomplish our purpose we particularize (1) by setting

a

⁢

(

z

)

≡

0

and

b

⁢

(

z

)

=

-

z

. Thus (1) becomes

d

⁢

ζ

d

⁢

z

+

ζ

2

=

z

.

(2)

(2) can be reduced to a

linear equation

of the second order by the suitable change:

ζ

=

w

′

⁢

(

z

)

/

w

⁢

(

z

)

, whence

ζ

′

=

w

′′

w

-

w

′

⁣

2

w

2

,

ζ

2

=

(

w

′

w

)

2

,

which leads (2) to

w

′′

-

z

⁢

w

=

0

.

(3)

Pairs of

linearly independent

solutions of (3) are the Airy functions.

Title

connection between Riccati equation and Airy functions

Canonical name

ConnectionBetweenRiccatiEquationAndAiryFunctions

Date of creation

2013-03-22 18:09:07

Last modified on

2013-03-22 18:09:07

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

5

Author

perucho (2192)

Entry type

Derivation

Classification

msc 35-00

Classification

msc 34-00

Generated on Fri Feb 9 20:43:09 2018 by

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dependence on initial conditions of solutions of ordinary differential equations

dependence on initial conditions of solutions of ordinary differential equations

Let

E

⊂

W

where

W

is a

normed vector space

,

f

∈

C

1

⁢

(

E

)

is a

continuous

differentiable map

f

:

E

→

W

. Furthermore consider the

ordinary differential equation

x

˙

=

f

⁢

(

x

)

with the

initial condition

x

⁢

(

0

)

=

x

0

.

Let

x

⁢

(

t

)

be the solution of the above

initial value problem

defined as

x

:

I

→

E

where

I

=

[

-

a

,

a

]

. Then there exist

δ

>

0

such that for all

y

0

∈

N

δ

⁢

(

x

0

)

(

y

0

in the

δ

neighborhood

of

x

0

)

has a unique solution

y

⁢

(

t

)

to the initial value problem above except for the initial value changed to

x

⁢

(

0

)

=

y

0

. In

addition

y

⁢

(

t

)

is twice continouously differentialble

function

of

t

over the

interval

I

.

Title

dependence on initial conditions of solutions of ordinary differential equations

Canonical name

DependenceOnInitialConditionsOfSolutionsOfOrdinaryDifferentialEquations

Date of creation

2013-03-22 13:37:19

Last modified on

2013-03-22 13:37:19

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

7

Author

Daume (40)

Entry type

Theorem

Classification

msc 35-00

Classification

msc 34-00

Generated on Fri Feb 9 20:43:11 2018 by

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existence and uniqueness of solution of ordinary differential equations

existence and uniqueness of solution of ordinary differential equations

Let

E

⊂

W

where

E

is an open subset of

W

which is a

normed vector space

, and let

f

be a

continuous

differentiable map

f

:

E

→

W

.

Then the

ordinary differential equation

defined as

x

˙

=

f

⁢

(

x

)

with the

initial condition

x

⁢

(

0

)

=

x

0

where

x

0

∈

E

has a unique solution on some interval containing zero. More specifically there exists

α

>

0

such that the following is a unique function

x

:

(

-

α

,

α

)

→

E

such that

x

˙

=

f

∘

x

and

x

⁢

(

0

)

=

x

0

.

[

HS

]

References

HS

Hirsch, W. Morris, Smale, Stephen.:

Differential Equations

,

Dynamical Systems

, And Linear Algebra. Academic Press, Inc. New York, 1974.

Title

existence and uniqueness of solution of ordinary differential equations

Canonical name

ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations

Date of creation

2013-03-22 13:36:50

Last modified on

2013-03-22 13:36:50

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

13

Author

Daume (40)

Entry type

Theorem

Classification

msc 35-00

Classification

msc 34-00

Classification

msc 34A12

Related topic

PicardsTheorem2

Related topic

CauchyKowalewskiTheorem

Generated on Fri Feb 9 20:43:15 2018 by

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index of differential equations

index of differential equations

Differential equations for functions on

ℝ

1.

General solution

, particular solutions,

singular solution

2.

Equation  y” = f(x) (

http://planetmath.org/EquationYFx

)

3.

Separation of variables

4.

ODE types reductible to the variables separable case

5.

Exact differential equation

6.

Linear differential equation of first order

7.

Second order linear differential equation with constant coefficients

8.

General solution of linear differential equation

9.

Frobenius method

10.

Bessel’s equation

11.

Chebyshev equation

12.

Hermite equation

13.

van der Pol equation

14.

Hypergeometric equation

15.

Laguerre’s

differential equation

(

http://planetmath.org/LaguerrePolynomial

)

16.

Second order ordinary differential equation

17.

Bernoulli equation

18.

d’Alembert’s equation (

http://planetmath.org/DAlembertsEquation

)

19.

Clairaut’s equation

20.

Riccati equation

Ordinary differential equations

1.

Linear ordinary differential equations

2.

Differential equation of the

isogonal trajectories

, the

tilt curves

3.

Frenet equations

4.

Hamilton equations

(

http://planetmath.org/HamiltonianEquations

)

5.

Geodesic equation

6.

Gradient system

7.

Lorenz equation

Partial differential equations

1.

Gradient system

2.

Euler-Lagrange equations (

http://planetmath.org/EulerLagrangeDifferentialEquation

)

3.

Hamilton equations

4.

Heat equation

5.

Laplace equation

6.

Poisson’s equation

7.

Wave equation

8.

d’Alembert and D. Bernoulli solutions of wave equation (

http://planetmath.org/DAlembertAndDBernoulliSolutionsOfWaveEquation

)

9.

Helmholtz equation

10.

Maxwell’s equations

11.

Korteweg - de Vries equation

12.

Schrödinger’s wave equation

13.

Dirac equation

14.

Klein-Gordon equation

15.

Hamilton-Jacobi equation

16.

Telegraph equation

17.

Black-Scholes equation

18.

Solved examples:

a) example of solving the heat equation

b)

time-dependent example of heat equation

c) using

Laplace transform

to solve heat equation

d) solving the wave equation due to D. Bernoulli (

http://planetmath.org/SolvingTheWaveEquationByDBernoulli

)

Complex

1.

Beltrami differential equation

2.

Cauchy-Riemann equations

Title

index of differential equations

Canonical name

IndexOfDifferentialEquations

Date of creation

2013-03-22 15:14:44

Last modified on

2013-03-22 15:14:44

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

39

Author

matte (1858)

Entry type

Topic

Classification

msc 35-00

Classification

msc 34-00

Related topic

ClairautsEquation

Generated on Fri Feb 9 20:43:17 2018 by

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Cauchy-Kowalewski theorem

Cauchy-Kowalewski theorem

Consider a system of

partial differential equations

involving

m

dependent variables

u

1

,

…

,

u

m

and

n

+

1

independent variables

t

,

x

1

,

…

,

x

n

:

∂

⁡

u

i

∂

⁡

t

=

F

i

⁢

(

u

1

,

…

,

u

m

;

t

,

x

1

,

…

,

x

n

;

∂

⁡

u

1

∂

⁡

x

1

,

…

,

∂

⁡

u

m

∂

⁡

x

n

)

in which

F

1

,

…

,

F

m

are

analytic functions

in a neighborhood of a point

(

u

1

0

,

…

,

u

m

0

;

t

0

,

x

1

0

,

…

,

x

n

0

)

subject to the

boundary
conditions

u

i

=

f

i

⁢

(

x

1

,

…

,

x

n

)

when

t

=

t

0

for given

functions

f

i

which are

analytic

in a neighborhood of

x

1

0

,

…

,

x

n

0

such that

u

i

0

=

f

i

⁢

(

x

1

0

,

…

,

x

n

0

)

.

The

Cauchy-Kowalewski theorem

asserts that this

boundary value problem

has a unique analytic solution

u

i

=

f

i

⁢

(

t

,

x

1

,

…

,

x

n

)

in a neighborhood of

(

u

1

0

,

…

,

u

m

0

;

0

,

x

1

0

,

…

,

x

n

0

)

.

The Cauchy-Kowalewski theorem is a local

existence theorem

— it only asserts that a solution exists in a neighborhood of the point, not in all space. A peculiar feature of this

theorem

is that the type of the

differential equation

(whether it is elliptic, parabolic, or hyperbolic) is irrelevant. As soon as one either considers global solutions or relaxes the

assumption

of analyticity, this is no longer the case — the

existence

and uniqueness of solutions to a differential equation (or system of differential equations) will depend upon the type of the equation.

By simple

transformations

, one can generalize this theorem.

By making a change of variable

t

′

=

t

-

ϕ

⁢

(

x

1

,

…

,

x

m

)

, with

ϕ

analytic, one can generalize the theorem to the case where the boundary values are specified on a surface given by the equation

t

=

ϕ

⁢

(

x

1

,

…

,

x

m

)

rather than on the plane

t

=

t

0

.

One can allow

higher order derivatives

by the device of introducing new variables. For

instance

, to allow third order time derivatives of

u

1

, one could introduce new variables

u

1

⁢

t

and

u

1

⁢

t

⁢

t

and augment the system of equations by adding

∂

⁡

u

1

∂

⁡

t

=

u

1

⁢

t

and

∂

⁡

u

1

⁢

t

∂

⁡

t

=

u

1

⁢

t

⁢

t

Likewise, one can eliminate spatial

derivatives

. The manner in which one introduces new equations for these new variables is somewhat clumsy to describe in general, and it is best to explain it by example, as is done in a

http://planetmath.org/node/6198

supplement to this entry.

Title

Cauchy-Kowalewski theorem

Canonical name

CauchyKowalewskiTheorem

Date of creation

2013-03-22 14:37:04

Last modified on

2013-03-22 14:37:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

16

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 35A10

Synonym

Cauchy-Kovalevskaya theorem

Related topic

Analytic

Related topic

CauchyInitialValueProblem

Related topic

ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations

Generated on Fri Feb 9 20:43:23 2018 by

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Holmgren uniqueness theorem

Holmgren uniqueness theorem

Given a system of linear

partial differential equations

with

analytic

coefficients

a

j

⁢

i

1

,

…

,

i

m

and

b

i

∑

j

,

i

1

,

…

,

i

m

a

j

⁢

i

1

,

…

,

i

m

⁢

(

x

1

,

…

,

x

m

)

⁢

∂

i

1

+

⋯

+

i

m

⁡

u

j

∂

⁡

x

1

i

1

⁢

…

⁢

∂

⁡

x

m

i

m

=

b

i

⁢

(

x

1

,

…

,

x

m

)

and analytic Cauchy data specified along a noncharacteristic analytic surface, there exists a neighborhood of the surface such that every smooth solution of the system defined in that neighborhood is analytic.

This theorem stengthens the

Cauchy-Kowalewski theorem

. While the latter theorem asserts that a unique analytic solution exists, it still allows the possibility that there might exist non-analytic solutions. Holmgren’s theorem asserts that this is not the case for linear systems.

It is often possible to determine the neighborhood in which Holmgren’s theorem holds explicitly. For instance, for many hyperbolic equations, one can show that this neighborhood can be taken to be the entire domain of dependence of the surface along which the boundary values were specified.

Title

Holmgren uniqueness theorem

Canonical name

HolmgrenUniquenessTheorem

Date of creation

2013-03-22 14:37:24

Last modified on

2013-03-22 14:37:24

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 35A10

Generated on Fri Feb 9 20:43:27 2018 by

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smooth linear partial differential equation without solution

smooth linear partial differential equation without solution

Cauchy-Kowalewski theorem

says that real analytic

partial differential equations

with real analytic initial data always have solutions. On the other hand Hans Lewy showed in 1957 that this is not true if the equation is only smooth. The example is obvious once we have the following theorem.

Theorem

(Lewy).

Let

x

,

y

,

z

be independent real variables. Let

f

be a

C

1

real function.
Suppose that there exists a

C

1

solution

u

to the following equation

[

-

∂

∂

⁡

x

-

i

⁢

∂

∂

⁡

y

+

2

⁢

i

⁢

(

x

+

i

⁢

y

)

⁢

∂

∂

⁡

z

]

⁢

u

=

f

′

⁢

(

z

)

,

in some neighbourhood of a point

(

0

,

0

,

z

0

)

.

Then

f

is real analytic at

z

0

.

Hence we need only pick

f

which is smooth and not real analytic at

z

0

and we have an example. For example,
let

z

0

=

0

and

f

⁢

(

x

)

=

∫

0

x

e

-

1

/

t

⁢

𝑑

t

.

References

1

Lewy, Hans.

Ann. of Math. (2)

66

(1957), 155–158.

Title

smooth linear partial differential equation without solution

Canonical name

SmoothLinearPartialDifferentialEquationWithoutSolution

Date of creation

2013-03-22 17:39:38

Last modified on

2013-03-22 17:39:38

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Example

Classification

msc 35A10

Classification

msc 35A05

Generated on Fri Feb 9 20:43:29 2018 by

LaTeXML
Green’s function

Green’s function

Some general preliminary considerations

Let

(

Ω

,

μ

)

be a

bounded

measure space and

ℱ

⁢

(

Ω

)

be a

linear function

space of

bounded functions

defined on

Ω

, i.e.

ℱ

⁢

(

Ω

)

⊂

ℒ

∞

⁢

(

Ω

)

.
We would like to note two types of

functionals

from the

dual space

(

ℱ

⁢

(

Ω

)

)

*

, which
will be used here:

1.

Each function

g

⁢

(

x

)

∈

ℒ

1

⁢

(

Ω

)

defines a functional

φ

∈

(

ℱ

⁢

(

Ω

)

)

*

in the
following way:

φ

⁢

(

f

)

=

∫

Ω

g

⁢

(

x

)

⁢

f

⁢

(

x

)

⁢

𝑑

μ

.

Such functional we will call

regular

functional and function

g

— its

generator

.

2.

For each

x

∈

Ω

, we will consider a functional

δ

x

∈

(

ℱ

⁢

(

Ω

)

)

*

defined as follows:

δ

x

⁢

(

f

)

=

f

⁢

(

x

)

.

(1)

Since generally, we can not speak about values at the point for functions from

(

L

)

∞

,
in the following, we assume some regularity for functions from considered spaces, so that
(

1

) is correctly defined.

Necessary notations and motivation

Let

(

Ω

x

,

μ

x

)

,

(

Ω

y

,

μ

y

)

be some bounded measure spaces;

ℱ

⁢

(

Ω

x

)

,

𝒢

⁢

(

Ω

y

)

be some
linear function spaces. Let

A

:

ℱ

⁢

(

Ω

x

)

→

𝒢

⁢

(

Ω

y

)

be a

linear operator

which has a well-defined

inverse

A

-

1

:

𝒢

⁢

(

Ω

y

)

→

ℱ

⁢

(

Ω

x

)

.

Consider an

operator

equation:

A

⁢

f

=

g

(2)

where

f

∈

ℱ

⁢

(

Ω

x

)

is unknown and

g

∈

𝒢

⁢

(

Ω

y

)

is given. We are interested to have an integral representation
for solution of (

2

). For this purpose we write:

f

⁢

(

x

)

=

δ

x

⁢

(

f

)

=

δ

x

⁢

(

A

-

1

⁢

(

g

)

)

=

[

(

A

-

1

)

*

⁢

δ

x

]

⁢

(

g

)

.

Definition of Green’s function

If

∀

x

∈

Ω

x

the functional

(

A

-

1

)

*

⁢

δ

x

is regular with generator

G

⁢

(

⋅

,

y

)

∈

ℒ

1

⁢

(

Ω

y

)

, then

G

is called

Green’s function of
operator

A

and solution of (

2

) admits the following integral representation:

f

⁢

(

x

)

=

∫

Ω

y

G

⁢

(

x

,

y

)

⁢

g

⁢

(

y

)

⁢

𝑑

μ

y

Title

Green’s function

Canonical name

GreensFunction

Date of creation

2013-03-22 14:43:36

Last modified on

2013-03-22 14:43:36

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 35C15

Related topic

PoissonsEquation

Generated on Fri Feb 9 20:43:32 2018 by

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initial value problem

initial value problem

Consider the

differential equation

d

⁢

y

d

⁢

x

=

x

.

The solution goes by writing

d

⁢

y

=

x

⁢

d

⁢

x

and then integrating both sides as

∫

𝑑

y

=

∫

x

⁢

𝑑

x

. The solution becomes

y

=

x

2

/

2

+

C

where

C

is any constant.

Differentiating

x

2

/

2

+

5

,

x

2

/

2

+

7

and some other examples shows that all these functions satisfy the condition imposed by the differential equation. So we have an infinite number of solutions.

An

initial value problem

is then a differential equation (ordinary or partial, or even a system) which, besides of stating the relation among the derivatives, also, by giving the

initial conditions

, specifies the values of the unknowns at some point.  This makes it possible to get a unique solution from the infinite number of ones.  However, it should be noted that not all possible choices of initial conditions specify a unique solution. It is also possible that there will be more than one solution satisfying a particular choice of initial conditions, or perhaps no solution. In fact, one of the main problems in the theory of differential equations is to determine what sort of initial conditions will single out a unique solution of a given differential equation.

In the

partial differential equations

of physics containing spatial variables (

x

etc.) and time variable (

t

), often one is seeking a solution within a certain spatial domain.  The conditions set for the solution of the equation may be called the

boundary conditions

when they determine the behavior of the solution function in the boundary (points) of the domain (or in the infinity); the other conditions concerning the solution function in some given moments of time are the proper

initial conditions

.

In our example we could add the initial condition

y

⁢

(

4

)

=

3

turning it into an initial value problem.  The

general solution

y

⁢

(

x

)

:=

x

2

/

2

+

C

is then restricted by

4

2

2

+

C

=

3

and by solving for

C

we obtain

C

=

-

5

, and so the unique solution for the system

d

⁢

y

d

⁢

x

=

x

y

⁢

(

4

)

=

3

is

y

⁢

(

x

)

=

x

2

/

2

-

5

.

Another way to consider an initial condition is seen in context of the tractrix.

Title

initial value problem

Canonical name

InitialValueProblem

Date of creation

2013-03-22 14:03:17

Last modified on

2013-03-22 14:03:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 35E15

Classification

msc 34A12

Related topic

CauchyInitialValueProblem

Defines

initial condition

Defines

boundary condition

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Beltrami differential equation

Beltrami differential equation

Suppose that

μ

:

G

⊂

ℂ

→

ℂ

is a

measurable function

, then the

partial differential equation

f

z

¯

⁢

(

z

)

=

μ

⁢

(

z

)

⁢

f

z

⁢

(

z

)

is called the

Beltrami differential equation

.

If furthermore

|

μ

⁢

(

z

)

|

<

1

and
in fact

|

μ

⁢

(

z

)

|

has a uniform bound less then 1 over the domain of definition, then the solution is a

quasiconformal mapping

with

complex

dilation (

http://planetmath.org/QuasiconformalMapping

)

μ

⁢

(

z

)

and maximal small dilatation (

http://planetmath.org/QuasiconformalMapping

)

d

f

=

sup

z

⁡

|

μ

⁢

(

z

)

|

.

A

conformal mapping

has

f

z

¯

≡

0

and so the solution can be

conformal

if and only if

μ

≡

0

.

The

partial derivatives

f

z

and

f

z

¯

(where

z

¯

is the

complex conjugate

of

z

) can here be given in terms of
the real and imaginary parts of

f

=

u

+

i

⁢

v

as

f

z

=

1

2

⁢

(

u

x

+

v

y

)

+

i

2

⁢

(

v

x

-

u

y

)

,

f

z

¯

=

1

2

⁢

(

u

x

-

v

y

)

+

i

2

⁢

(

v

x

+

u

y

)

.

References

1

L. V. Ahlfors.

. Van Nostrand-Reinhold, Princeton, New Jersey, 1966

Title

Beltrami differential equation

Canonical name

BeltramiDifferentialEquation

Date of creation

2013-03-22 14:08:34

Last modified on

2013-03-22 14:08:34

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 35F20

Classification

msc 30C62

Related topic

QuasiconformalMapping

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