This is a numerical example of an elliptic integral of the type I(-e₅-e₆-e₇-e₈). The integral is:

ó õ x y     1 Ö   P(t)    Q(t)  dt

P(t)=  70620 t4+76505 t3+55212 t2-3960 t-47080 70620

Q(t)=  1056 t4-253 t3+256 t2-768 t-1232 1056

pr contains the roots of P(t) to 64 digits.

pr=[0.6183750362285898611630954367509939371735341569814494139652073224,

-0.9606793347322929762516912462599391563300795580130657769564559538,

-0.3705145174148151091223687619121940570883939661508584851710423510

+i 0.9924413678822316252059880839221510776586955523504581104933012564,

-0.3705145174148151091223687619121940570883939661508584851710423510

-i 0.9924413678822316252059880839221510776586955523504581104933012564]

qr contains the roots of Q(t) to 64 digits.

qr=[1.205317170123685215133793366969311476533755786975102250194410814,

-0.7706035515514024607008469009359815872325305200392761918270488486,

-0.09756514261947471054980656634999827798394596680124636251701431669

+i 1.116491276187300608884347679081910543311602803814869927467250770,

-0.09756514261947471054980656634999827798394596680124636251701431669

-i 1.116491276187300608884347679081910543311602803814869927467250770]

The limits of integration y and x are between branch points. y=[ 14/51]    x=[ 25/78].

Now define the integral in terms we can use to reduce it to basic integrals.

m=[-e5-e6-e7-e8]       m=[0,0,0,0,-1,-1,-1,-1]

a=-append(pr,qr)

b=[1,1,1,1,1,1,1,1]

Do partial fraction expansion using function D219(i,m,n,a,b,e).

D219(1,m,8,a,b,e)=

I(-e5) 0.1719034337591826067998940155243381247710120540484288203408232048

-I(-e6) 0.2977835907816666108684642146910620770656933045746933535741304899

+I(-e7) (0.06294007851124200203428509958336197614734062526313226661665364254

+i 0.1900551068072961448452971696636458994778588750417174698968007894)

+I(-e8) (0.06294007851124200203428509958336197614734062526313226661665364254

-i 0.1900551068072961448452971696636458994778588750417174698968007894)

I(-e₅), I(-e₆), I(-e₇), and I(-e₈) are basic integrals.

Look up I(-e₅) in table using function IS(v,h).

IS([IA(-e5)],4)=[I427(a,b,x,y)]

Calculate I(-e₅) using function I427(a,b,x,y).

m=[0,0,0,0,-1]     a=-append(pr,[qr1])     x=  25 78     y=  14 51

Change the sign of a₁ to correct for the square root of negative values. Change the sign of b₁ also. Change the sign of a₅ to correct for negative values. Change the sign of b₅ also.

a1=pr1     a5=qr1     b=[-1,1,1,1,-1]

I(-e5)=i I427(a,b,x,y)

I(-e5)=i 0.06672191073158007107623026369228937278082546730266834155066753759

Calculate I(-e₆) using function I427(a,b,x,y).

m=[0,0,0,0,-1]     a=-append(pr,[qr2])     x=  25 78     y=  14 51

Change the sign of a₁ to correct for the square root of negative values. Change the sign of b₁ also.

a1=pr1     b=[-1,1,1,1,1]

I(-e6)=  I427(a,b,x,y) i

I(-e6)=-i 0.05669033539449556294808519644607854857375753077650818500839662123

Calculate I(-e₇) using function I427(a,b,x,y).

m=[0,0,0,0,-1]     a=-append(pr,[qr3])     x=  25 78     y=  14 51

Change the sign of a₁ to correct for the square root of negative values. Change the sign of b₁ also.

a1=pr1     b=[-1,1,1,1,1]

I(-e7)=  I427(a,b,x,y) i

I(-e7)=0.04819023820058985209585591229021823367818642606701558142696003546

-i 0.01705338473599866759270810605057820713270071630785338759369707557

Calculate I(-e₈) using function I427(a,b,x,y).

m=[0,0,0,0,-1]     a=-append(pr,[qr4])     x=  25 78     y=  14 51

Change the sign of a₁ to correct for the square root of negative values. Change the sign of b₁ also.

a1=pr1     b=[-1,1,1,1,1]

I(-e8)=  I427(a,b,x,y) i

I(-e8)=-0.04819023820058985209585591229021823367818642606701558142696003546

-i 0.01705338473599866759270810605057820713270071630785338759369707557

Substitute in the values of I(-e₅), I(-e₆), I(-e₇), and I(-e₈).

I(m)=i 0.04452209618635411720490440524633702835452503987783510229967900266

Do numerical integration find a check value.

ó õ x y     1 Ö   P(t)    Q(t)  dt=

i 0.04452209618635411720490440524633702835452503987783510229967900268

The results match.

Jim FitzSimons

Mailto:cherry@neta.com