This is a numerical example of an elliptic integral of the type $I(-e_5-e_6-e_7-e_8)$. The integral is:

${\int }_y^x\hspace{1em}\frac{1}{\sqrt{P(t)}\hspace{1em}Q(t)}\hspace{1em}\mathrm{dt}$

$P(t)=\frac{70620\hspace{1em}{t}^4+76505\hspace{1em}{t}^3+55212\hspace{1em}{t}^2-3960\hspace{1em}t-47080}{70620}$

$Q(t)=\frac{1056\hspace{1em}{t}^4-253\hspace{1em}{t}^3+256\hspace{1em}{t}^2-768\hspace{1em}t-1232}{1056}$

$\mathrm{pr}$ contains the roots of P(t) to 64 digits.

$\mathrm{pr}=[0.6183750362285898611630954367509939371735341569814494139652073224$,

$-0.9606793347322929762516912462599391563300795580130657769564559538$,

$-0.3705145174148151091223687619121940570883939661508584851710423510$

$\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+i\hspace{1em}0.9924413678822316252059880839221510776586955523504581104933012564$,

$-0.3705145174148151091223687619121940570883939661508584851710423510$

$\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-i\hspace{1em}0.9924413678822316252059880839221510776586955523504581104933012564]$

$\mathrm{qr}$ contains the roots of Q(t) to 64 digits.

$\mathrm{qr}=[1.205317170123685215133793366969311476533755786975102250194410814$,

$-0.7706035515514024607008469009359815872325305200392761918270488486$,

$-0.09756514261947471054980656634999827798394596680124636251701431669$

$\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+i\hspace{1em}1.116491276187300608884347679081910543311602803814869927467250770$,

$-0.09756514261947471054980656634999827798394596680124636251701431669$

$\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-i\hspace{1em}1.116491276187300608884347679081910543311602803814869927467250770]$

The limits of integration y and x are between branch points. $y=\frac{14}{51}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=\frac{25}{78}$.

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

$m=[-e_5-e_6-e_7-e_8]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}m=[0,0,0,0,-1,-1,-1,-1]$

$a=-\mathrm{append}(\mathrm{pr},\mathrm{qr})$

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

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

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

$I(-e_5)\hspace{1em}0.1719034337591826067998940155243381247710120540484288203408232048$

$-I(-e_6)\hspace{1em}0.2977835907816666108684642146910620770656933045746933535741304899$

$+I(-e_7)\hspace{1em}(0.06294007851124200203428509958336197614734062526313226661665364254$

$+i\hspace{1em}0.1900551068072961448452971696636458994778588750417174698968007894)$

$+I(-e_8)\hspace{1em}(0.06294007851124200203428509958336197614734062526313226661665364254$

$-i\hspace{1em}0.1900551068072961448452971696636458994778588750417174698968007894)$

$I(-e_5)$, $I(-e_6)$, $I(-e_7)$, and $I(-e_8)$ are basic integrals.

Look up $I(-e_5)$ in table using function $\mathrm{IS}(v,h)$.

$\mathrm{IS}([\mathrm{IA}(-e5)],4)=[I427(a,b,x,y)]$

Calculate $I(-e_5)$ using function $I427(a,b,x,y)$.

$m=[0,0,0,0,-1]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}a=-\mathrm{append}(\mathrm{pr},[{\mathrm{qr}}_1])\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=\frac{25}{78}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=\frac{14}{51}$

Change the sign of $a_1$ to correct for the square root of negative values. Change the sign of $b_1$ also. Change the sign of $a_5$ to correct for negative values. Change the sign of $b_5$ also.

$a_1={\mathrm{pr}}_1\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{a}_5={\mathrm{qr}}_1\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}b=[-1,1,1,1,-1]$

$I(-e_5)=i\hspace{1em}I427(a,b,x,y)$

$I(-e_5)=i\hspace{1em}0.06672191073158007107623026369228937278082546730266834155066753759$

Calculate $I(-e_6)$ using function $I427(a,b,x,y)$.

$m=[0,0,0,0,-1]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}a=-\mathrm{append}(\mathrm{pr},[{\mathrm{qr}}_2])\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=\frac{25}{78}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=\frac{14}{51}$

Change the sign of $a_1$ to correct for the square root of negative values. Change the sign of $b_1$ also.

$a_1={\mathrm{pr}}_1\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}b=[-1,1,1,1,1]$

$I(-e_6)=\frac{I427(a,b,x,y)}{i}$

$I(-e_6)=-i\hspace{1em}0.05669033539449556294808519644607854857375753077650818500839662123$

Calculate $I(-e_7)$ using function $I427(a,b,x,y)$.

$m=[0,0,0,0,-1]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}a=-\mathrm{append}(\mathrm{pr},[{\mathrm{qr}}_3])\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=\frac{25}{78}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=\frac{14}{51}$

Change the sign of $a_1$ to correct for the square root of negative values. Change the sign of $b_1$ also.

$a_1={\mathrm{pr}}_1\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}b=[-1,1,1,1,1]$

$I(-e_7)=\frac{I427(a,b,x,y)}{i}$

$I(-e_7)=0.04819023820058985209585591229021823367818642606701558142696003546$

$\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-i\hspace{1em}0.01705338473599866759270810605057820713270071630785338759369707557$

Calculate $I(-e_8)$ using function $I427(a,b,x,y)$.

$m=[0,0,0,0,-1]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}a=-\mathrm{append}(\mathrm{pr},[{\mathrm{qr}}_4])\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}x=\frac{25}{78}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}y=\frac{14}{51}$

Change the sign of $a_1$ to correct for the square root of negative values. Change the sign of $b_1$ also.

$a_1={\mathrm{pr}}_1\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}b=[-1,1,1,1,1]$

$I(-e_8)=\frac{I427(a,b,x,y)}{i}$

$I(-e_8)=-0.04819023820058985209585591229021823367818642606701558142696003546$

$\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-i\hspace{1em}0.01705338473599866759270810605057820713270071630785338759369707557$

Substitute in the values of $I(-e_5)$, $I(-e_6)$, $I(-e_7)$, and $I(-e_8)$.

$I(m)=i\hspace{1em}0.04452209618635411720490440524633702835452503987783510229967900266$

Do numerical integration find a check value.

${\int }_y^x\hspace{1em}\frac{1}{\sqrt{P(t)}\hspace{1em}Q(t)}\hspace{1em}\mathrm{dt}=$

$i\hspace{1em}0.04452209618635411720490440524633702835452503987783510229967900268$

The results match.

Jim FitzSimons

Mailto:cherry@neta.com