reduction of elliptic integrals to standard form

reduction of elliptic integrals to standard form

Any

integral

of the form

∫

R

⁢

(

x

,

P

⁢

(

x

)

)

⁢

𝑑

x

, where

R

is a

rational
function

and

P

is a

polynomial

of degree 3 or 4 can be expressed as a linear
combination of

elementary functions

and

elliptic integrals

of the first, second,
and

third kinds

.

To begin, we will assume that

P

has no repeated roots. Were this not the case,
we could simply pull the repeated factor out of the

radical

and be left with a
polynomial of degree of 1 or 2 inside the

square root

and express the integral in
terms of

inverse trigonometric functions

.

Make a change of variables

z

=

(

a

⁢

x

+

b

)

/

(

c

⁢

x

+

d

)

. By choosing the coefficients

a

,

b

,

c

,

d

suitably, one can cast P into either Jacobi’s normal form

P

⁢

(

z

)

=

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

or Weierstrass’ normal form

P

⁢

(

z

)

=

4

⁢

z

3

-

g

2

⁢

z

-

g

3

.

Note that

R

⁢

(

z

,

P

⁢

(

z

)

)

=

A

⁢

(

z

)

+

B

⁢

(

z

)

⁢

P

⁢

(

z

)

C

⁢

(

z

)

+

D

⁢

(

z

)

⁢

P

⁢

(

z

)

for suitable polynomials

A

,

B

,

C

,

D

. We can rationalize the

denominator

like so:

A

⁢

(

z

)

+

B

⁢

(

z

)

⁢

P

⁢

(

z

)

C

⁢

(

z

)

+

D

⁢

(

z

)

⁢

P

⁢

(

z

)

×

C

⁢

(

z

)

-

D

⁢

(

z

)

⁢

P

⁢

(

z

)

C

⁢

(

z

)

-

D

⁢

(

z

)

⁢

P

⁢

(

z

)

=

F

⁢

(

z

)

+

G

⁢

(

z

)

⁢

P

⁢

(

z

)

The rational functions

F

and

G

appearing in the foregoing equation are defined like so:

F

⁢

(

z

)

=

A

⁢

(

z

)

⁢

C

⁢

(

z

)

-

B

⁢

(

z

)

⁢

D

⁢

(

z

)

⁢

P

⁢

(

z

)

C

2

⁢

(

z

)

-

D

2

⁢

(

z

)

⁢

P

⁢

(

z

)

G

⁢

(

z

)

=

2

⁢

B

⁢

(

z

)

⁢

C

⁢

(

z

)

-

A

⁢

(

z

)

⁢

D

⁢

(

z

)

C

2

⁢

(

z

)

-

D

2

⁢

(

z

)

⁢

P

⁢

(

z

)

Since

∫

F

⁢

(

z

)

⁢

𝑑

z

may be expressed in terms of elementary functions,
we shall focus our attention on the remaining piece,

∫

G

⁢

(

z

)

⁢

P

⁢

(

z

)

⁢

𝑑

z

,
which we shall write as

∫

H

⁢

(

z

)

/

P

⁢

(

z

)

⁢

𝑑

z

, where

H

=

P

⁢

G

..
Because we may decompose

H

into

partial fractions

, it suffices to consider
the following cases, which we shall all

A

n

and

B

n

:

A

n

⁢

(

z

)

=

∫

z

n

P

⁢

(

z

)

⁢

𝑑

z

B

n

⁢

(

z

,

r

)

=

∫

1

(

z

-

r

)

n

⁢

P

⁢

(

z

)

⁢

𝑑

z

Here,

n

is a non-negative

integer

and

r

is a

complex number

.

We will reduce thes further using

integration by parts

.
Taking

antiderivatives

, we have:

∫

z

n

-

1

⁢

(

z

⁢

P

′

⁢

(

z

)

+

2

⁢

n

⁢

P

⁢

(

z

)

)

2

⁢

P

⁢

(

z

)

⁢

𝑑

z

=

z

n

⁢

P

⁢

(

z

)

+

C

∫

(

z

-

r

)

⁢

P

′

⁢

(

z

)

-

2

⁢

n

⁢

P

⁢

(

z

)

2

⁢

(

z

-

r

)

n

+

1

⁢

P

⁢

(

z

)

⁢

𝑑

z

=

P

⁢

(

z

)

(

z

-

r

)

n

+

C

These

identities

will allow us to express

A

n

’s and

B

n

’s with large

n

in terms of ones with smaller

n

’s.

At this point, it is convenient to employ the specific form of the polynominal

P

.
We will first conside the Weierstrass normal form and then the Jacobi normal form.

Substituting into our identities and collecting terms, we find

4

⁢

(

2

⁢

n

+

3

)

⁢

A

n

+

2

=

(

2

⁢

n

+

1

)

⁢

g

2

⁢

A

n

+

2

⁢

n

⁢

g

3

⁢

A

n

-

1

+

z

n

⁢

4

⁢

z

3

-

g

2

⁢

x

-

g

3

+

C

2

⁢

n

⁢

(

4

⁢

r

3

-

g

2

⁢

r

-

g

3

)

⁢

B

n

+

1

+

(

2

⁢

n

-

1

)

⁢

(

12

⁢

r

2

-

g

2

)

⁢

B

n

+

24

⁢

(

n

-

1

)

⁢

r

⁢

B

n

-

1

+

4

⁢

(

2

⁢

n

-

3

)

⁢

B

n

-

2

+

4

⁢

z

3

-

g

2

⁢

x

-

g

3

(

z

-

r

)

n

+

C

=

0

Note that there are some cases which can be integrated in elementary terms. Namely, suppose that the power is odd:

∫

z

2

⁢

m

+

1

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

𝑑

z

Then we may make a change of variables

y

=

z

2

to obtain

1

2

⁢

∫

y

2

⁢

m

⁢

(

1

-

y

)

⁢

(

1

-

k

2

⁢

y

)

⁢

𝑑

y

,

which may be integrated using elementary functions.

Next, we derive some identities using integration by parts. Since

d

⁢

(

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

)

=

(

9

2

⁢

k

2

⁢

z

3

-

3

⁢

(

1

+

k

2

)

⁢

z

)

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

d

⁢

z

,

we have

(

2

⁢

m

+

1

)

∫

z

2

⁢

m

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

𝑑

z

+

∫

z

2

⁢

m

+

1

⁢

(

9

2

⁢

k

2

⁢

z

3

-

3

⁢

(

1

+

k

2

)

⁢

z

)

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

𝑑

z

=

z

2

⁢

m

+

1

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

+

C

By colecting terms, this identity may be rewritten as follows:

(

1

+

2

⁢

m

+

9

2

⁢

k

2

)

∫

z

2

⁢

m

+

4

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

𝑑

z

-

(

4

+

2

⁢

m

)

⁢

(

1

+

k

2

)

∫

z

2

⁢

m

+

2

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

𝑑

z

+

∫

z

2

⁢

m

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

=

x

2

⁢

k

+

1

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

⁢

(

1

-

z

2

)

⁢

(

1

-

k

2

⁢

z

2

)

+

C

By repeated use of this identity, we may express any integral of the form

∫

z

2

⁢

m

⁢

P

⁢

(

z

)

⁢

𝑑

z

as the sum of a linear combination of

∫

z

2

⁢

P

⁢

(

z

)

⁢

𝑑

z

and

∫

P

⁢

(

z

)

⁢

𝑑

z

and the product of a polyomial and

P

⁢

(

z

)

.

Likewise, we can use integration by parts to simplify integrals of the form

∫

P

⁢

(

z

)

(

z

-

r

)

n

⁢

𝑑

z

Will finish later — saving in case of computer crash.

Title

reduction of elliptic integrals to standard form

Canonical name

ReductionOfEllipticIntegralsToStandardForm

Date of creation

2014-02-01 18:13:38

Last modified on

2014-02-01 18:13:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

30

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 33E05

Related topic

ExpressibleInClosedForm

Generated on Fri Feb 9 20:37:52 2018 by

LaTeXML