Reduction tables for elliptic integrals

Jim FitzSimons

July 28, 2000

If you do not have the symbol fonts on your UNIX system you can get the L^AT_EX source for this index. L^AT_EX source

This is the general form of the equations in these tables.

I(m) = ó õ x y  h Õ i=1  (ai +bi t)-1/2 n Õ j=1  (aj +bj t)mj dt ,

(1)

h=3 or 4, and the m's are integers. We assume that the integral is well defined (in particular that the open interval of integration contains no finite branch point of the integrand) and that no two of the n linear factors on the right side are proportional. The a's and b's, some of which may be complex, can be left as symbols. Here m=(m₁,...,m_n)=åⁿ_j=1 m_j ej , where ej is an n-tuple with 1 in the jth place and 0's elsewhere. For example, if h=n=3 then I(e1-e3)=I(1,0,-1).

A(m) denotes the value of the integrand of I(m) at t=x minus its value at t=y.

A(m) = h Õ i=1  (ai +bi x)-1/2 n Õ j=1  (aj +bj x)mj - h Õ i=1  (ai +bi y)-1/2 n Õ j=1  (aj +bj y)mj .

(2)

dij means d_ij and is defined as:

dij = ai bj - aj bi .

(3)

The a's are replaced by d's in these tables.

Here is the table of integral reductions for h=3 and n=3. table h=3 n=3

Here is the table of integral reductions for h=3 and n=4. table h=3 n=4

Here is the table of integral reductions for h=4 and n=4. table h=4 n=4

Here is the table of integral reductions for h=4 and n=5. table h=4 n=5

This is simplified form of the equations in these tables.

~ I   (m) = ó õ x y  h Õ i=1  æ è  ai bi + t ö ø -1/2   n Õ j=1  æ è  aj bj + t ö ø mj    dt ,

(4)

~ A   (m) = 2  h Õ i=1  æ è  ai bi + x ö ø -1/2   n Õ j=1  æ è  aj bj + x ö ø mj   - 2  h Õ i=1  æ è  ai bi + y ö ø -1/2   n Õ j=1  æ è  aj bj + y ö ø mj   .

(5)

rij means r_ij and is defined as:

rij =  ai bi -  aj bj  .

(6)

The a's are replaced by r's in these tables.

Here is the table of integral reductions for h=3 and n=3. table h=3 n=3

Here is the table of integral reductions for h=3 and n=4. table h=3 n=4

Here is the table of integral reductions for h=4 and n=4. table h=4 n=4

Here is the table of integral reductions for h=4 and n=5. table h=4 n=5

This is the new notation of the equations in these tables.

^ I   (m) = I(m)/B,     ^ A   (m) = A(m)/B,     B = n Õ j=1  bjmj.

(7)

Here is the table of integral reductions for h=3 and n=3. table h=3 n=3

Here is the table of integral reductions for h=3 and n=4. table h=3 n=4

Here is the table of integral reductions for h=4 and n=4. table h=4 n=4

Here is the table of integral reductions for h=4 and n=5. table h=4 n=5

C1998.MTH Utilities for the reduction of symmetric elliptic integrals.

c1999.mth More utilities for symmetric elliptic integrals.

c2000.mth More utilities for symmetric elliptic integrals.

CARLSON.DMO A DEMO for numerical evaluation of symmetric elliptic integrals.

CARLSON.DOC A text document describing how to do the numerical evaluation of symmetric elliptic integrals.

carlson.mth DERIVE program to do the numerical evaluation of symmetric elliptic integrals.

Carlson.txt A text document describing how to do the numerical evaluation of symmetric elliptic integrals.

elex.dmo A DERIVE example using symmetric elliptic integrals to solve a problem.

elex.mth A DERIVE example using symmetric elliptic integrals to solve a problem.

lenbez.dmo A DERIVE example solving the length of a Bezier curve using symmetric elliptic integrals.

lenbez.mth A DERIVE example solving the length of a Bezier curve using symmetric elliptic integrals.

lenbez5.dmo A DERIVE version 5 example solving the length of a Bezier curve using symmetric elliptic integrals.

lenbez5.mth A DERIVE version 5 example solving the length of a Bezier curve using symmetric elliptic integrals.

PPEX.DMO A DERIVE example solving an integral from particle physics using symmetric elliptic integrals.

ppex.mth A DERIVE example solving an integral from particle physics using symmetric elliptic integrals.

TABLE.BAS A BASIC program to convert the DERIVE table of integrals to a more usable form.

table.dmo A DERIVE program to generate a table of integrals.

table.mth A DERIVE program to generate a table of integrals.

TABLE.TXT A table of integrals.

TABLER.BAS A BASIC program to convert the DERIVE table of integrals to a more usable form.

tabler.dmo A DERIVE program to generate a table of integrals.

tabler.mth A DERIVE program to generate a table of integrals.

TABLER.TXT A table of integrals.

c2001.mth Utility to generate tables for symmetric elliptic integrals.

TABLEN.BAS A BASIC program to convert the DERIVE table of integrals to a more usable form.

tablen.dmo A DERIVE program to generate a table of integrals.

tablen.mth A DERIVE program to generate a table of integrals.

TABLEN.TXT A table of integrals.

lenbezn.dmo A DERIVE example solving the length of a Bezier curve using symmetric elliptic integrals.

lenbezn.mth A DERIVE example solving the length of a Bezier curve using symmetric elliptic integrals.

c2001a.mth More utilities for symmetric elliptic integrals.

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