aarjaneiro/planetmath huggingface dataset:
possible orders of elliptic functions

possible orders of elliptic functions

The order of a non-trivial

elliptic function

cannot be zero. This is a
simple consequence of Liouville’s theorem. Were the order of an elliptic
function zero, then the

function

would have no poles. By definition,
an elliptic function has no

essential singularities

and is

doubly
periodic

. Hence, if the degree were zero, the function would be

continuous

everywhere and hence, being doubly periodic, would be

bounded

(since

continuous functions

on a

compact

domain (like the
closure of the

fundamental parallelogram

) are bounded). By Liouville’s
theorem, this would imply that the function is constant.

The order of an elliptic function cannot be 1. This follows from the
fact that the

residues

at the poles of an elliptic function within a
fundamental parallelogram must sum to zero — if the function were of
degree 1, it would have exactly one first-order pole in the
fundamental parallelgram but any first-order pole must have a non-zero
residue.

Any number

greater

than one is possible as the order of an elliptic
function. As an example of an elliptic function of order two, we may
take the Weierstass

℘

-function, which has a single pole of order 2
in the fundamental domain. The

n

-th

derivative

of this function
will have a single pole of order

n

+

2

in the fundamental domain,
hence be of order

n

+

2

, so is an example showing that, for every

integer

greater than 2, there exists an elliptic function having that
integer as its order.

Title

possible

orders of elliptic functions

Canonical name

PossibleOrdersOfEllipticFunctions

Date of creation

2013-03-22 15:48:30

Last modified on

2013-03-22 15:48:30

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33E05

Related topic

PeriodicFunctions

Generated on Fri Feb 9 20:37:43 2018 by

LaTeXML
product representations of Jacobi ϑ functions

product representations of Jacobi

ϑ

functions

The

Jacobi theta functions

can be expressed as infinite products:

ϑ

1

⁢

(

z

;

q

)

=

2

⁢

q

1

/

4

⁢

sin

⁡

z

⁢

∏

n

=

1

∞

(

1

-

q

2

⁢

n

)

⁢

(

1

-

2

⁢

q

2

⁢

n

⁢

cos

⁡

2

⁢

z

+

q

4

⁢

n

)

ϑ

2

⁢

(

z

;

q

)

=

2

⁢

q

1

/

4

⁢

cos

⁡

z

⁢

∏

n

=

1

∞

(

1

-

q

2

⁢

n

)

⁢

(

1

+

2

⁢

q

2

⁢

n

⁢

cos

⁡

2

⁢

z

+

q

4

⁢

n

)

ϑ

3

⁢

(

z

;

q

)

=

∏

n

=

1

∞

(

1

-

q

2

⁢

n

)

⁢

(

1

+

2

⁢

q

2

⁢

n

-

1

⁢

cos

⁡

2

⁢

z

+

q

4

⁢

n

-

2

)

ϑ

4

⁢

(

z

;

q

)

=

∏

n

=

1

∞

(

1

-

q

2

⁢

n

)

⁢

(

1

-

2

⁢

q

2

⁢

n

-

1

⁢

cos

⁡

2

⁢

z

+

q

4

⁢

n

-

2

)

Title

product representations of Jacobi

ϑ

functions

Canonical name

ProductRepresentationsOfJacobivarthetaFunctions

Date of creation

2013-03-22 14:52:13

Last modified on

2013-03-22 14:52:13

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33E05

Generated on Fri Feb 9 20:37:44 2018 by

LaTeXML
proof of Jacobi’s identity for ϑ functions

proof of Jacobi’s identity for

ϑ

functions

We start with the

Fourier transform

of

f

⁢

(

x

)

=

e

i

⁢

π

⁢

τ

⁢

x

2

+

2

⁢

i

⁢

x

⁢

z

:

∫

-

∞

+

∞

e

i

⁢

π

⁢

τ

⁢

x

2

+

2

⁢

i

⁢

x

⁢

z

⁢

e

2

⁢

π

⁢

i

⁢

x

⁢

y

⁢

𝑑

x

=

(

-

i

⁢

τ

)

-

1

/

2

⁢

e

-

i

⁢

(

z

+

π

⁢

y

)

2

π

⁢

τ

Applying the

Poisson summation formula

, we obtain the following:

∑

n

=

-

∞

+

∞

e

i

⁢

π

⁢

τ

⁢

n

2

+

2

⁢

i

⁢

n

⁢

z

=

(

-

i

⁢

τ

)

-

1

/

2

⁢

∑

n

=

-

∞

+

∞

e

-

i

⁢

(

z

+

π

⁢

n

)

2

π

⁢

τ

The left hand equals

ϑ

3

(

z

∣

τ

)

. The right hand can be rewritten as follows:

∑

n

=

-

∞

+

∞

e

-

i

⁢

(

z

+

π

⁢

n

)

2

π

⁢

τ

=

e

-

i

⁢

z

2

π

⁢

τ

∑

n

=

-

∞

+

∞

e

-

i

⁢

π

⁢

n

2

τ

-

2

⁢

i

⁢

n

⁢

z

τ

=

e

-

i

⁢

z

2

π

⁢

τ

ϑ

3

(

z

/

τ

∣

-

1

/

τ

)

Combining the two expressions yields

ϑ

3

(

z

∣

τ

)

=

e

-

i

⁢

z

2

π

⁢

τ

ϑ

3

(

z

/

τ

∣

-

1

/

τ

)

Title

proof of Jacobi’s identity for

ϑ

functions

Canonical name

ProofOfJacobisIdentityForvarthetaFunctions

Date of creation

2013-03-22 14:47:01

Last modified on

2013-03-22 14:47:01

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

19

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 33E05

Generated on Fri Feb 9 20:37:47 2018 by

LaTeXML
Weierstrass sigma function

Weierstrass sigma function

Definition 1

.

Let

Λ

⊂

C

be a

lattice

. Let

Λ

∗

denote

Λ

-

{

0

}

.

1.

The

Weierstrass sigma function

is defined as the
product

σ

⁢

(

z

;

Λ

)

=

z

⁢

∏

w

∈

Λ

∗

(

1

-

z

w

)

⁢

e

z

/

w

+

1

2

⁢

(

z

/

w

)

2

2.

The

Weierstrass zeta function

is defined by the sum

ζ

⁢

(

z

;

Λ

)

=

σ

′

⁢

(

z

;

Λ

)

σ

⁢

(

z

;

Λ

)

=

1

z

+

∑

w

∈

Λ

∗

(

1

z

-

w

+

1

w

+

z

w

2

)

Note that the Weierstrass zeta function is basically the
derivative of the

logarithm

of the sigma function. The

zeta
function

can be rewritten as:

ζ

⁢

(

z

;

Λ

)

=

1

z

-

∑

k

=

1

∞

𝒢

2

⁢

k

+

2

⁢

(

Λ

)

⁢

z

2

⁢

k

+

1

where

𝒢

2

⁢

k

+

2

is the

Eisenstein series

of weight

2

⁢

k

+

2

.

3.

The

Weierstrass eta function

is defined to be

η

⁢

(

w

;

Λ

)

=

ζ

⁢

(

z

+

w

;

Λ

)

-

ζ

⁢

(

z

;

Λ

)

,

for
any

⁢

z

∈

ℂ

(It can be proved that this is well defined,
i.e.

ζ

⁢

(

z

+

w

;

Λ

)

-

ζ

⁢

(

z

;

Λ

)

only depends on

w

).
The Weierstrass eta function must not be confused with the

Dedekind eta function

.

Title

Weierstrass sigma function

Canonical

name

WeierstrassSigmaFunction

Date of creation

2013-03-22 13:54:06

Last modified on

2013-03-22 13:54:06

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 33E05

Synonym

sigma function

Synonym

zeta function

Synonym

eta function

Related topic

EllipticFunction

Related topic

ModularDiscriminant

Defines

Weierstrass sigma function

Defines

Weierstrass zeta function

Defines

Weierstrass eta function

Generated on Fri Feb 9 20:37:55 2018 by

LaTeXML
℘-function

℘

-function

Let

L

be a lattice on

ℂ

. The Weierstrass

℘

function

associated to

L

is given by

℘

⁢

(

z

)

=

1

z

2

+

∑

w

∈

L

∖

{

0

}

(

1

(

z

-

w

)

2

-

1

w

2

)

.

The

℘

function is

meromorphic

and

analytic

on

ℂ

∖

L

, whereas at each

w

∈

L

, it has an order

2

pole. It is also an

even function

, because

℘

⁢

(

z

)

=

℘

⁢

(

-

z

)

.

Its

derivative

℘

′

⁢

(

z

)

=

-

2

⁢

∑

w

∈

L

1

(

z

-

w

)

3

is also an odd, meromorphic, and

elliptic function

, analytic at

ℂ

∖

L

and having order

3

poles at each

w

∈

L

.

The functions

℘

and

℘

′

form together a generator set for the field of elliptic functions associated to the lattice

L

.

Title

℘

-function

Canonical name

wpfunction

Date of creation

2013-03-22 14:51:54

Last modified on

2013-03-22 14:51:54

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 33E05

Synonym

℘

Synonym

Weierstrass

℘

function

Synonym

Weierstrass p-function

Synonym

p-Weierstrass

Synonym

Weierstrass

℘

-function

Related topic

EllipticCurve

Related topic

EllipticFunction

Related topic

ExamplesOfEllipticFunctions

Generated on Fri Feb 9 20:37:57 2018 by

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Mittag-Leffler function

Mittag-Leffler function

The

Mittag-Leffler function

E

α

⁢

β

is a

complex function

which depends on two

complex

parameters

α

and

β

. It may be defined by the following series when the

real part

of

α

is

strictly

positive

:

E

α

⁢

β

⁢

(

z

)

=

∑

k

=

0

∞

z

k

Γ

⁢

(

α

⁢

k

+

β

)

In this case, the series converges for all values of the argument

z

, so the Mittag-Leffler function is an

entire function

.

Title

Mittag-Leffler function

Canonical

name

MittagLefflerFunction

Date of creation

2013-03-22 14:54:34

Last modified on

2013-03-22 14:54:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 33E12

Generated on Fri Feb 9 20:37:59 2018 by

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generating function of Hermite polynomials

generating function of Hermite polynomials

We start from the definition of

Hermite polynomials

via their

http://planetmath.org/node/11983

Rodrigues formula

H

n

(

z

)

:=

(

-

1

)

n

e

z

2

d

n

d

⁢

z

n

e

-

z

2

(

n

=

0

,

1

,

2

,

…

)

.

(1)

The consequence

f

(

n

)

⁢

(

z

)

=

n

!

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

ζ

)

(

ζ

-

z

)

n

+

1

⁢

𝑑

ζ

(2)

of

http://planetmath.org/node/1150

Cauchy integral formula

allows to write (1) as the

complex integral

H

n

⁢

(

z

)

=

(

-

1

)

n

⁢

n

!

2

⁢

i

⁢

π

⁢

∮

C

e

z

2

-

ζ

2

(

ζ

-

z

)

n

+

1

⁢

𝑑

ζ

,

where

C

is any contour around the point

z

and the direction is anticlockwise.  The

http://planetmath.org/node/11373

substitution

z

-

ζ

:=

t

here yields

H

n

⁢

(

z

)

=

n

!

2

⁢

i

⁢

π

⁢

∮

C

′

e

z

2

-

(

z

-

t

)

2

t

n

+

1

⁢

𝑑

t

,

where the contour

C

′

goes round the origin.  Accordingly, by (2) we can infer that

H

n

⁢

(

z

)

=

[

d

n

d

⁢

t

n

⁢

e

z

2

-

(

z

-

t

)

2

]

t

=

0

,

whence we have found the generating function

e

z

2

-

(

z

-

t

)

2

=

∑

n

=

0

∞

H

n

⁢

(

z

)

⁢

t

n

n

!

of the Hermite polynomials.

Title

generating function of Hermite polynomials

Canonical name

GeneratingFunctionOfHermitePolynomials

Date of creation

2013-03-22 19:05:25

Last modified on

2013-03-22 19:05:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Derivation

Classification

msc 33E30

Classification

msc 33B99

Classification

msc 26C05

Classification

msc 26A09

Classification

msc 12D99

Related topic

OrthogonalPolynomials

Related topic

ExampleOfFindingTheGeneratingFunction

Related topic

GeneratingFunctionOfLaguerrePolynomials

Related topic

VariantOfCauchyIntegralFormula

Generated on Fri Feb 9 20:38:02 2018 by

LaTeXML
symbolic computation

symbolic computation

1

Symbolic Computation

Also called

formula manipulation

or

algebraic computation

.

Symbolic computation

refers to the automatic

transformation

of mathematical expressions in symbolic form, hence in an exact way, as opposed to numerical and hence limited-precision floating-point

computation

. Typical

operations

include differentiation and integration,

linear algebra

and matrix calculus, operations with

polynomials

, or the simplification of algebraic expressions.

Programs

or systems in this area which provide a

language

interface are called

Computer Algebra Systems

(or CASes). There are also symbolic computation libraries for existing programming languages.

Primarily designed for applications in theoretical physics or mathematics, these systems (which are often interactive in the case of CASes) can be used in any area where straightforward but tedious or lengthy calculations with formulae are required.

2

Systems

Some well known, general symbolic computation CASes are:

•

http://www.axiom-developer.org

Axiom

•

http://www.scientek.com/macsyma/mxmain.htm

Macsyma

•

http://maxima.sourceforge.net/

GNU Maxima

•

http://www.maplesoft.com/

products

/maple/

Maple

•

http://www.wolfram.com/products/

mathematica

/index.html

Mathematica

•

http://www.uni-koeln.de/REDUCE/

Reduce

These systems have different scope and facilities, and some are easier to use or to access than others. There is a trend away from generalized CAS systems to more specialized, application-specific systems, such as:

•

http://www.

singular

.uni-kl.de/

SINGULAR (

algebraic geometry

, esp. singular

varieties

)

•

http://pari.math.u-bordeaux.fr/

PARI-GP (computations on curves)

•

http://www-gap.dcs.st-and.ac.uk/ gap/

GAP (group theory)

Some non-CAS symbolic computation libraries, with their supported languages, are:

•

http://www.ginac.de/

GiNaC (C++)

References

1

Based on content from the

http://rkb.home.cern.ch/rkb/titleA.html

Data Analysis Briefbook

Title

symbolic computation

Canonical name

SymbolicComputation

Date of creation

2013-03-22 12:04:20

Last modified on

2013-03-22 12:04:20

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

18

Author

akrowne (2)

Entry type

Topic

Classification

msc 33F99

Classification

msc 17-08

Classification

msc 16Z05

Classification

msc 13P99

Classification

msc 12Y05

Classification

msc 11Y40

Classification

msc 14Q99

Classification

msc 68W30

Synonym

formula manipulation

Synonym

algebraic computation

Defines

CAS

Defines

computer algebra systems

Generated on Fri Feb 9 20:38:09 2018 by

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all solution of the Lorenz equation enter an ellipsoid

all solution of the Lorenz equation enter an ellipsoid

If

σ

,

τ

,

β

>

0

then all solutions of the

Lorenz equation

x

˙

=

σ

⁢

(

y

-

x

)

y

˙

=

x

⁢

(

τ

-

z

)

-

y

z

˙

=

x

⁢

y

-

β

⁢

z

will enter an ellipsoid centered at

(

0

,

0

,

2

⁢

τ

)

in finite time.
In

addition

the solution will remain inside the ellipsoid once it
has entered. To observe this we define a

Lyapunov function

V

⁢

(

x

,

y

,

z

)

=

τ

⁢

x

2

+

σ

⁢

y

2

+

σ

⁢

(

z

-

2

⁢

τ

)

2

.

It then
follows that

V

˙

=

2

⁢

τ

⁢

x

⁢

x

˙

+

2

⁢

σ

⁢

y

⁢

y

˙

+

2

⁢

σ

⁢

(

z

-

2

⁢

τ

)

⁢

z

˙

=

2

⁢

τ

⁢

x

⁢

σ

⁢

(

y

-

x

)

+

2

⁢

σ

⁢

y

⁢

(

x

⁢

(

τ

-

z

)

-

y

)

+

2

⁢

σ

⁢

(

z

-

2

⁢

τ

)

⁢

(

x

⁢

y

-

β

⁢

z

)

=

-

2

⁢

σ

⁢

(

τ

⁢

x

2

+

y

2

+

β

⁢

(

z

-

r

)

2

-

b

⁢

τ

2

)

.

We then choose an ellipsoid which all the solutions will enter and
remain inside. This is done by choosing a constant

C

>

0

such that
the ellipsoid

τ

⁢

x

2

+

y

2

+

β

⁢

(

z

-

r

)

2

=

b

⁢

τ

2

is

strictly

contained in the ellipsoid

τ

⁢

x

2

+

σ

⁢

y

2

+

σ

⁢

(

z

-

2

⁢

τ

)

2

=

C

.

Therefore all solution will eventually enter and remain inside the above ellipsoid since

V

˙

<

0

when a solution is located at the exterior of the
ellipsoid.

Title

all solution of the Lorenz equation enter an ellipsoid

Canonical

name

AllSolutionOfTheLorenzEquationEnterAnEllipsoid

Date of creation

2013-03-22 15:15:28

Last modified on

2013-03-22 15:15:28

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

4

Author

Daume (40)

Entry type

Result

Classification

msc 34-00

Classification

msc 65P20

Classification

msc 65P30

Classification

msc 65P40

Classification

msc 65P99

Generated on Fri Feb 9 20:38:25 2018 by

LaTeXML
bibliography for ordinary differential equations

bibliography for ordinary differential equations

Reference for ordinary differential equations

General theory,

34A

[A

]

[HS

]

Qualitative theory

34E

[C

]

[GH

]

[P

]

Online books

[T

]

Teschl, Gerald

,

http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html

http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html, 2004.

Title

bibliography for ordinary differential equations

Canonical name

BibliographyForOrdinaryDifferentialEquations

Date of creation

2013-03-22 15:19:01

Last modified on

2013-03-22 15:19:01

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

11

Author

Daume (40)

Entry type

Bibliography

Classification

msc 34-00

Generated on Fri Feb 9 20:38:27 2018 by

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contact point

contact point

Let

ϕ

be a

planar dynamical system

and

γ

a curve. Then a

contact point

of the curve and the

dynamical system

occurs when the tangent vector of the curve is collinear to the vector of the dynamical system.

[

KGA

]

References

KGA

Khovanskiĭ, A.G.: Fewnomials.

American Mathematical Society

, Providence, 1991.

Title

contact point

Canonical name

ContactPoint

Date of creation

2013-03-22 14:06:22

Last modified on

2013-03-22 14:06:22

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 34-00

Classification

msc 34A99

Generated on Fri Feb 9 20:38:29 2018 by

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dynamical system

dynamical system

A

dynamical system

on

X

where

X

is an open subset of

ℝ

n

is a differentiable map

ϕ

:

ℝ

×

X

→

X

where

ϕ

⁢

(

t

,

𝐱

)

=

ϕ

t

⁢

(

𝐱

)

satisfies

i

ϕ

0

⁢

(

𝐱

)

=

𝐱

for all

𝐱

∈

X

(the identity function)

ii

ϕ

t

∘

ϕ

s

⁢

(

𝐱

)

=

ϕ

t

+

s

⁢

(

𝐱

)

for all

s

,

t

∈

ℝ

(composition)

[

HSD

]

[

PL

]

Note that a

planar dynamical system

is the same definition as above but with

X

an open subset of

ℝ

2

.

References

HSD

Hirsch W. Morris, Smale, Stephen, Devaney L. Robert:

Differential Equations

, Dynamical Systems & An Introduction to Chaos

(Second Edition)

. Elsevier Academic Press, New York, 2004.

PL

Perko, Lawrence: Differential Equations and Dynamical Systems

(Third Edition)

. Springer, New York, 2001.

Title

dynamical system

Canonical name

DynamicalSystem

Date of creation

2013-03-22 14:06:25

Last modified on

2013-03-22 14:06:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

14

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 34-00

Classification

msc 37-00

Synonym

supercategorical dynamics

Related topic

SystemDefinitions

Related topic

GroupoidCDynamicalSystem

Related topic

CategoricalDynamics

Related topic

Bifurcation

Related topic

ChaoticDynamicalSystem

Related topic

IndexOfCategories

Defines

planar dynamical system

Generated on Fri Feb 9 20:38:32 2018 by

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Lienard system

Lienard system

A

Lienard system

is a planar

ordinary differential equation

x

˙

=

y

-

f

⁢

(

x

)

y

˙

=

-

g

⁢

(

x

)

with conditions on the smoothness of

f

and

g

. It is equivalent
to the following

second order ordinary differential equation

x

¨

+

f

′

⁢

(

x

)

⁢

x

˙

+

g

⁢

(

x

)

=

0

.

Example:

•

van der Pol equation

References

P

Perko, Lawrence

,

Differential Equations

and

Dynamical Systems

,
Springer, New York, 2001.

Title

Lienard system

Canonical

name

LienardSystem

Date of creation

2013-03-22 15:20:45

Last modified on

2013-03-22 15:20:45

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Definition

Classification

msc 34-00

Defines

Lienard equation

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

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linear ordinary differential equation

linear ordinary differential equation

The following problem is a

linear ordinary differential equation

:

Let

A

:

I

→

ℂ

n

×

n

be a known matrix. Find
a matrix

x

:

I

→

ℂ

n

×

n

(or vector

x

:

I

→

ℂ

n

) such that

x

′

=

A

⁢

x

,

x

⁢

(

0

)

=

x

0

,

where

x

0

is a known initial value.

Title

linear ordinary differential equation

Canonical

name

LinearOrdinaryDifferentialEquation

Date of creation

2013-03-22 15:31:11

Last modified on

2013-03-22 15:31:11

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

4

Author

matte (1858)

Entry type

Definition

Classification

msc 34-00

Classification

msc 35-00

Generated on Fri Feb 9 20:38:40 2018 by

LaTeXML
Lyapunov function

Lyapunov function

Suppose we are given an

autonomous system

of first order

differential equations

.

d

⁢

x

d

⁢

t

=

F

⁢

(

x

,

y

)

d

⁢

y

d

⁢

t

=

G

⁢

(

x

,

y

)

Let the origin be an isolated

critical point

of the above system.

A

function

V

⁢

(

x

,

y

)

that is of class

C

1

and satisfies

V

⁢

(

0

,

0

)

=

0

is called a

Lyapunov function

if every open ball

B

δ

⁢

(

0

,

0

)

contains at least one point where

V

>

0

.

If
there happens to exist

δ

*

such that the function

V

˙

, given by

V

˙

⁢

(

x

,

y

)

=

V

x

⁢

(

x

,

y

)

⁢

F

⁢

(

x

,

y

)

+

V

y

⁢

(

x

,

y

)

⁢

G

⁢

(

x

,

y

)

is

positive definite

in

B

δ

*

⁢

(

0

,

0

)

. Then the origin is
an

unstable

critical point of the system.

Title

Lyapunov function

Canonical name

LyapunovFunction

Date of creation

2013-03-22 13:42:29

Last modified on

2013-03-22 13:42:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 34-00

Synonym

Liapunov function

Generated on Fri Feb 9 20:38:46 2018 by

LaTeXML
method of undetermined coefficients

method of undetermined coefficients

Given a (usually non-homogeneous)

ordinary differential equation

F

⁢

(

x

,

f

⁢

(

x

)

,

f

′

⁢

(

x

)

,

…

,

f

(

n

)

⁢

(

x

)

)

=

0

,

the

method of undetermined coefficients

is a way of finding an exact solution when a guess can be made as to the general form of the solution.

In this method, the form of the solution is guessed with unknown

coefficients

left as

variables

. A typical guess might be of the form

A

⁢

e

2

⁢

x

or

A

⁢

x

2

+

B

⁢

x

+

C

. This can then be substituted into the

differential equation

and solved for the coefficients. Obviously the method requires knowing the approximate form of the solution, but for many problems this is a feasible requirement.

This method is most commonly used when the formula is some combination of

exponentials

, polynomials,

sin

and

cos

.

Example

Suppose we have the following second order non-homogeneous equation

f

′′

⁢

(

x

)

-

2

⁢

f

′

⁢

(

x

)

+

f

⁢

(

x

)

=

2

⁢

e

2

⁢

x

.

If we guess that the soution is of the form

f

⁢

(

x

)

=

A

⁢

e

2

⁢

x

, then, by substitution, we get

4

⁢

A

⁢

e

2

⁢

x

-

4

⁢

A

⁢

e

2

⁢

x

+

A

⁢

e

2

⁢

x

-

2

⁢

e

2

⁢

x

=

0

and therefore

A

⁢

e

2

⁢

x

=

2

⁢

e

2

⁢

x

, so

A

=

2

, giving

f

⁢

(

x

)

=

2

⁢

e

2

⁢

x

as a solution.

Title

method of undetermined coefficients

Canonical

name

MethodOfUndeterminedCoefficients

Date of creation

2013-03-22 12:52:55

Last modified on

2013-03-22 12:52:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 34-00

Related topic

ODE

Related topic

DifferentialEquation

Generated on Fri Feb 9 20:38:48 2018 by

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methods of solving ODEs

methods of solving ODEs

There are many different methods of solving

ordinary differential equations

. These are tabulated here and divided into sections for linear and non-linear ODEs. These sections are further subdivided into exact, numerical and quantitative methods.

1

Linear ODEs

1.1

Exact methods

•

General solution of linear differential equation

•

Linear differential equation of first order

•

Method of

separation of variables

•

Method of integrating factors

•

Method of undetermined coefficients

•

Frobenius’ Method

•

Method of Green’s functions

1.2

Numerical methods

•

Runge-Kutta method

•

Heun method

•

Rayleigh-Ritz method

1.3

Quantitative methods

2

Non-linear ODEs

Title

methods of solving ODEs

Canonical name

MethodsOfSolvingODEs

Date of creation

2013-03-22 17:52:02

Last modified on

2013-03-22 17:52:02

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

9

Author

invisiblerhino (19637)

Entry type

Topic

Classification

msc 34-00

Generated on Fri Feb 9 20:38:50 2018 by

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