elliptic integrals and Jacobi elliptic functions

elliptic integrals and Jacobi elliptic functions

Elliptic integrals

For a

modulus

0

<

k

<

1

(while here, we define the

complementary modulus

to

k

to be the positive number

k

′

with

k

2

+

k

′

⁣

2

=

1

) , write

F

⁢

(

ϕ

,

k

)

=

∫

0

ϕ

d

⁢

θ

1

-

k

2

⁢

sin

2

⁡

θ

(1)

E

⁢

(

ϕ

,

k

)

=

∫

0

ϕ

1

-

k

2

⁢

sin

2

⁡

θ

⁢

𝑑

θ

(2)

Π

⁢

(

n

,

ϕ

,

k

)

=

∫

0

ϕ

d

⁢

θ

(

1

+

n

⁢

sin

2

⁡

θ

)

⁢

1

-

k

2

⁢

sin

2

⁡

θ

(3)

The change of variable

x

=

sin

⁡

ϕ

turns these into

F

1

⁢

(

x

,

k

)

=

∫

0

x

d

⁢

v

(

1

-

v

2

)

⁢

(

1

-

k

2

⁢

v

2

)

(4)

E

1

⁢

(

x

,

k

)

=

∫

0

x

1

-

k

2

⁢

v

2

1

-

v

2

⁢

𝑑

v

(5)

Π

1

⁢

(

n

,

x

,

k

)

=

∫

0

x

d

⁢

v

(

1

+

n

⁢

v

2

)

⁢

(

1

-

v

2

)

⁢

(

1

-

k

2

⁢

v

2

)

(6)

The first three

functions

are known as Legendre’s form of the

incomplete
elliptic integrals

of the first, second, and

third kinds

respectively.
Notice that (2) is the special case

n

=

0

of (3).
The latter three are known as Jacobi’s form of those

integrals

.
If

ϕ

=

π

/

2

, or

x

=

1

, they are called

complete

rather than incomplete
integrals, and we refer to the auxiliary elliptic integrals

K

⁢

(

k

)

=

F

⁢

(

π

/

2

,

k

)

,

E

⁢

(

k

)

=

E

⁢

(

π

/

2

,

k

)

, etc.

One use for elliptic integrals is to systematize the evaluation of
certain other integrals.
In particular, let

p

be a third- or fourth-degree

polynomial

in one variable, and let

y

=

p

⁢

(

x

)

.
If

q

and

r

are any two polynomials in two variables, then the

indefinite integral

∫

q

⁢

(

x

,

y

)

r

⁢

(

x

,

y

)

⁢

𝑑

x

has a “

closed form

” in terms of the above incomplete elliptic integrals,
together with

elementary functions

and their

inverses

.

Jacobi’s

elliptic functions

In (1) we may regard

ϕ

as a function of

F

, or vice versa.
The notation used is

ϕ

=

am

⁢

u

u

=

arg

⁢

ϕ

and

ϕ

and

u

are known as the amplitude and

argument

respectively.
But

x

=

sin

⁡

ϕ

=

sin

⁡

am

⁢

u

.
The function

u

↦

sin

⁡

am

⁢

u

=

x

is denoted by

sn

and is one of four

Jacobi (or

Jacobian

)
elliptic functions

. The four are:

sn

⁢

u

=

x

cn

⁢

u

=

1

-

x

2

tn

⁢

u

=

sn

⁢

u

cn

⁢

u

dn

⁢

u

=

1

-

k

2

⁢

x

2

When the

Jacobian elliptic functions

are extended to

complex arguments

,
they are doubly periodic and have two poles in any parallelogram of
periods; both poles are simple.

Title

elliptic integrals and Jacobi elliptic functions

Canonical name

EllipticIntegralsAndJacobiEllipticFunctions

Date of creation

2013-03-22 13:58:28

Last modified on

2013-03-22 13:58:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 33E05

Related topic

ArithmeticGeometricMean

Related topic

PerimeterOfEllipse

Defines

elliptic integral

Defines

Jacobi elliptic function

Defines

Jacobian elliptic function

Defines

complementary modulus

Defines

complete elliptic integral

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