thompson. elliptical integrals

8/3/2019 Thompson. Elliptical Integrals.

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1 7 . Elliptic IntegralsL. M. MILNE-THOMEION

ContentsMathematical h p e r t i e a . . . . . . . . . . . . . . . . . . . .17.1. Definition of Elliptic Integrals . . . . . . . . . . . . .17.2. Canonical Forms . . . . . . . . . . . . . . . . . . .17.3. Complete Elliptic Integralsof the First and Second Kinds . .17.4. Incomplete Ellipt ic Int egralsof the First and Second Kinds .

17.5. Landen sTransformation . . . . . . . . . . . . . . . .17.6. The P r o c e s s of the Arithmetic-Geometric Mean . . . . . .17.7. Ellipt ic Int egrals of the Third Kind . . . . . . . . . . .Numerical Methods . . . . . . . . . . . . . . . : . . . . .17.8. Use and Extension of the Tables . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . .Table 17.1. Complete Ellipt ic Int egrals of the First and Second Kindsand the Nome q With Argument the Parameter m . . . . . . . . .

K(m),K W , 15D; d m ) , a(m) , 15D;W m ) ,E ( 4 , 9Dm= 0(.01) 1Table 17.2. Complete Ellipt ic Integrals of the Fixst and Second Kindsand the Nome pR it h Argument the Modular Angle a . . . . . .

K ( 4 , K (a),d4, q1(a),E b ) ,E M , 15Da= Oo( 0)9OoTable 17.3. ParametermWith Argument K (m )/ K (m ) . . . . . .

K (m )/ K (m ) 3(.02)3, 1ODTable 17.4. Auxiliary.Functions or Computation of the Nome Q and theParameterm . . . . . . . . . . . . . . . . . . . . . . . . .

Q( m) q i (m) i 15DL(m)=--K(m)+K*) T I n ($ lODml= 0(.01) . 15

Table 17.5. Ellipt ic Integral of the First Kind F(q\ a ) . . . . . . .~ ~= 0 ~ ( 2 ~ ) 9 0 ~ , (100)850,q= 0 (50)900, 8D

Table 17.6. Elliptic Integral of the Second Kind E(q\ a ) . . . . . .a= 0(20)900,5(100)850,q= 0 (50)900, 8D

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U n iv e r sit y of Arimna. ( P r e p a r e d u n d e r c o n t r a c t w i t h t h e N a t i o n a l B u r e a u ofBt.ndsrde.)

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http://132e971666309de6d1af6925d115c0f2.pdf/http://ac5a7318d227adfe8f2685dfb22e2a6.pdf/http://bb6586b2647ccf2d19ee62c618afe1f.pdf/http://132e971666309de6d1af6925d115c0f2.pdf/http://809ee067cdaa508dd92ee930aefe760.pdf/http://50379c4b779c6f564635212ba78271b4.pdf/http://bb6586b2647ccf2d19ee62c618afe1f.pdf/http://50379c4b779c6f564635212ba78271b4.pdf/http://809ee067cdaa508dd92ee930aefe760.pdf/http://132e971666309de6d1af6925d115c0f2.pdf/http://132e971666309de6d1af6925d115c0f2.pdf/http://ac5a7318d227adfe8f2685dfb22e2a6.pdf/http://132e971666309de6d1af6925d115c0f2.pdf/


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588 ELLIFTIC I N T E Q R A L SPsgeTable 17.7. Jacobian Zeta Function Z(q\ a) . . . . . . . . . . . . 619

Values of K(a)Z(q\ a)a= 0 (20)900,5 (100)850, 0 = 0~( 5~) 90~,D

Table 17.8. Heuman sLambda Function & (q\ a) . . . . . . . . . . 622

a= 0 (20)900, 5 (100)850,q= 0 (50)900Table 17.9. Elliptic Int egral of the Third Kind n ( n ;q\ a) . . . . . . 625

n= O(.1)1, q, a= 0(150)900, 5D

The author acknowledges with thanks the assistance of Ruth Zucker in the computa-tion of the examplea, Ruth E. Capuano for Table 17.3, David S. Liepman for Table 17.4,and A n d r e w Schopf for Table 17.9.
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58917 . lliptic In t eg ra l s

Mathematical Properties17.1. Delinition of Elliptic Integrals

If R(z, y) is a rational function of z and y,where$ s equal to a cubic or quar t ic polynomialin z, th e int egral1711 S R ( z , Wis called an elliptic integml.The elliptic integral just defined can not, ingeneral, be expressed in terms of elementaryfunctions.Exceptions to this are(i) when R(z, y) contains no odd powers of y.(ii) when t he polynomial$ as a repeated factor.

We therefore exclude these c-.By substitut ing for $ and d enoting by p,(z) apolynomial in z we get

where Rl(z) and R2(z) are r at ional functions of 2.Hence, by expreasing %( z ) as t he sum of a poly-nom ial and par tial fract ionsJ N Z , Y ) d z = ~ l ( Z ) d z + 2 s l , J 3 - L d z

+ B,B,J[ (Z-C)'Yl-'dzReduction FormulaeLet

1712$ = d + a lz " + 4 + a ~+ a 4 (I.ol+ a11# O)

= ~O ( Z -C ) ' + b! (z-c)'+ bz (x-c)'+ ~~(z-c) + 4(lbl+ l~l l# O)1713 I,=*y-'d;G, J , = [v(z-C ) I-'&S S

By integrating the derivatives of yz andy(z-c)-' we get the reduction formulae17.1.4( 8+ 2 )d ,+ 8 + 4 a i 3) I,+ 2+ a2(8+ 1 ) I, + i

+ 4 % ( 2? + l) I ,+ a a J , -l= Z Ly @=O, 1 , 2 , . . .)'See (17.71 Z4.72.

17.1.5(2-8)bJ,-a+ 4 b i (3-28)J,-z + b,( 1 -8)Js-l+ + a( 1-28)J,-8brJ .+ i= y(Z-C)-'

( a = & 2, 3 , . . .)By means of these reduction formulae and cer-tain transformations (see Examples 1 and 2)every elliptic integral can be brought to dependon t he int egral of a rat ional funct ion and on threecanon ical form s for ellipt ic integrals.

17.2. Canonical FormsDcdinitions1721

m= sina a ; m is the parameter ,1722 x= sin 9 = sn u1723 m cp= cn u

CY is the modular angle

1724(1-m sina cp))= dnu= A(cp), t he delt a am plitud e1725 cp= arcsin (en u)= am u, the amplitud eE l l i p t i c In- of the&t Kind1726 ~(cp\ a)F(qlm) = S1 -sins a sin2 e ) -W0

J od w = u= l

Elliptic In-1 of the Second Kind17.2.8 E(p\ a )= E(ulm)= ( I- ' ) -+ ( 1- t2)*dt17.2.10 = p n s w d W

17.2.11 = rn ,u+ m$~cn%dw


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590 ELLIPTIC I N T E G R A L S17.2.12 E(p\ a)= u-m sn2wdwL U

(For th eta funct ions, see chapter 16.)E l l i p t i c I n t e g r a l of t h e T h ir d K in d17.2.14

n ( n ;p \ a) =so (1 --n sin2 e) 1 [1-sin2 a sin2 e 1 - 1 ~I f z= sn ( u l m ) ,

17.2.15n ( n ;ul m) = I (l--nt2)- [(1-P ) (1 - mt2) ] - 1/ 2dt

17.2.16 = L( l- -n sn 2 (wlm))-ldwT h e A m p l i t u d e (p

17.2.17 p= am u= arcs in (an u)= arcs in zcan be calculated from Tables 17.5 and 4.14.

T h e P a r a m e t er mDependence on t he parameter m is denoted by avert ical stroke pr eceding t he param eter, e.g.,Together wi th the parameter we define theF(vlm).

complementary parameter ml by17.2.18 m+ m1= 1ar r snged , see 17.4, t h at O


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ELLIPTIC INTEGRALS 591R elat ion to th e H ypergeom etric Fu nct ion

(see chapter 15)17.3.9 K = 4*F O, 4;1; m)17.3.10 E= + r F(-+ , ;1; m)

Infinite Series17.3.11

17.3.12

Legendre sR elat ion17.3.13 EK + K-KK =

Auxiliary F un ction17.3.14 L(m)=K* In --K(m)6

r m117.3.15 m = 1- 6 exp [- r(K(m)+ L(m)) / K (m) ]17.3.16 m= 16 exp [-r(K (m) L(ml)) / K(m)]The function L(m) is tabulated in Table 17.4.

q-SeriesThe Nome p and the Complementary Nome p,

17.3.17 p= p(m)= exp [-rK / K]17.3.18 pl= p(ml) = exp [-rK/ K ]17.3.19 1 1In - n - 4. Q PI17.3.20

1 1!2 P1loglo- og,, -= (T log,, e) l= 1.86152 28349 to lOD

17.3.21

17.3.23

8 -117.3.24 am u = v + cD 2* sin 280where v= xu/ (2K)8 -1 8(1+ ?)

17.3.25 lhK E-K) = Om-nl17.3.26 lim [K-4 In (16/ m1)]= 0m+l17.3.27 lim m-,( K--E) = lh m -l(E-mnr ,K)= r/

Mo m 417.3.28 lim q/ m= lim pl/ m1= l/ 16

m 4 m r t lAlternative Evaluations of K and E (see also 17.5)

17.3.29K(m)= 2[1+ m: 2]- K([(1- 1+ rn: 2)l*)*

17.3.31 K(a)= 2F(wcttm ( s ~ c / ) \ a)17.3.32 & (a) 2E(anttan (secl )\ a)- +COS a

~o= 1.3862944 bo=.5a i= .11197 23 b1= .12134 78a = .07252 96 b2= .02887 29

(lo= 1.38629 4361 12 bo= .5a, = .09666 344259 b l = 12498 593597%= .03590 092383 ba= .06880 248576%= .03742 563713 b,= .03328 355346U4= .01451 196212 b4= .00441 787012

a The approximat ions17.3.33-17.3.36 are from C. H u ttinge, J r . , Approximations for Digital Computers, Prince-t on Univ. Prees. Princeton, N. J . (with parmission).* s e c p o g c 11.


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592 ELUPTIC

FIQUBE7.1. cbmpletedt?ipt!&~?Mk@d f the$I'dkind.

FIQUBB7.2. Complete 4ip t k integral of the8 d i n d .

I NTEGRALS17.3.35E(m)= [l + a1m1+ a&l+ [b1m,+ bm : ] In ( l lm l )

+ 4 m )(r(m)[


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where cota A is the posit ive root of the equat ionand m t an z p= tanz(p cot %- 1.17.4.12E((p+ $\ a) = E(X\ a).iE(p\ 90"-a)where

$-[cota (p+m s h h v ~ c a ~- m 1 1 5 - m 1 cot z~= o

b i + 2+ iF ( p \ 900-a )+ - b3bl= sinza sin X cos X sina ~((l-sin ' a sinaX)bx= (l-sina a sinz X ) ( l -W S a a sinap) ) sin p COS pb3= wsZp + si n a a sinz X sin' p

Amplitude Near to */ 2 ( s e e also 17.5)If cos a t an (p t an # = 1

17.4.13 F((p\ a)+ F (# \ a )=F ( r / 2\ a) K17.4.14

E((P\ a)+ E(# \ a)=E( r /Z\a) sinaa SiW S i n #Values when (p is near to r / 2and m is near to uni tycan be calculated by these formulae.

Parameter Greater Th an Un i t y17.4.15 F((pplm)= m-tF(@lm-l), sin e=m+ sin (p17.4.16 E(ulm) = m+ E (u m+ (m-i)- m- 1)uby which a parameter greater than uni ty can bereplaced by a parameter less than uni ty .17.4.17 Negative Par ameterF( 01 -m) = (1 + m) -+ K(m(+ m) - 1 )

-( l+ m ) -+ Fc-(pi m(l+ m)-l)17.4.18E(uI- ) = (1 + m)+ E ( u(1+ ) + l m ( m + I)-1)

-m(l+ m)-tsn( .u( l+ m)+ lm(l + m)-')cd(u(1 + m)'lm(l+ m!-l) 1

whereby comp ut at ions can be mad e for negativeparameters , and therefore for pure imaginarymodulus.F((P\ UI

FIGURE7.3. Incomplete d ip t ic integtal o the@8t kin&

F(o\ a), o conetentFIGWE 7.4. Incomplete d ip t ic ktegral of the# r st k i n d .

P((p\ a), u conatmnt


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594 ELLIPTIC I N T E G R A L S

FIGURE7.6. InCOmPkte elliptic ~?'&h?@df thes e e d kind.E(v \ U) , (p constant

spacial caoem17.4.19 F ( P \ O) = p11.4.20 F(i(p\O)= i( p

FIGURE7.7. Incomplete ellipt ic bt egral o theSeCond kind.E(v\ a), a constant

17.4.22 F(iP\ goO)= i arctan ( &h (p>17.4.23 E((P\ O) = P11.4.24 E(i(p\ O) i ( P17.4.25 E((p\ 90)= sin (p11.4.26 E(i(p\ 90)= i inh (p

FIGURE7.5. (p-90' F(9'a),- const a n t .K


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Jacmbi*oZeta Function17.4.!27 Z(v\ a)= E(v\ a)- (a)F(cp\ a)/ K(a)17.4.28 Z(ulm)= Z(u) = E@) -uE(m)/ K(m)17.4.29 Z(-u)= -Z(u)

Z(U+ 2K)= Z(U)7.4.3017.4.31 Z(K-U)= -Z(K+ u)

Heuman 's Lambda Fun ction17.4.39ao(v\ a)= F ( ~\ 9 0" -a )t(a) + 2-1K ( ~ ) z ( ~ \ w

2= - K(a)E((p\ W-a)n-17.4.40-[m a) -E(ol)lF(v\ 90O--a) 1


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83 ."c1

3

rnu

5+

E+


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E L L I P T I C INTEQRALS 597S o m e Impor tan t Special Gues

17.4.53

17.4.54

17.4.56

-( P

2 52'+ 11-2'l+ z'1--2z

0

45O

45O

4 5 O

45O

17.4.62

17.4.63

17.4.64

17.4.65" d t

17.4.66= d t17.4.67

C P17.4.68

17.4.69C P

Reduction of Sdt/ @ where P = P (t ) is a cubicpolynomial wit h t hree real factors P =(t--81) (t-a)t--8a) where 81>82>a. write17.4.61

R--&ml= cos ' a= -81--I%

cos (P

2 - 1 42 - 1 + 4 343+ 1- -2+ 1 + 24 3 - l+ z&+ 1-z1-43-21+ 2r3-2

0-1 5 O

1 5 O

75O

75O

Reduction of J &/ @ when P= P(t)= P+ alt*+ a d + % is a cubic polynomial with o n l y one reelroott= / 3. We form the first and second derivetivea P'(t), P"(t) with reape& to t and then write

1 1 P"@)17.4.70 X*=[P@]'n, m=&* a= ---2 8 [P '@)]"*17.4.71

17.4.72

17.4.73= d t

17.4.74 cu

17.5. Landen's TransformationM a n T d o r m a t i o n

Let an , a,,.+ lbe two modular a n gh such t hat17.5.1 ( l+ & u*~)(~+ cos a n )= 2 (a++lVn)

8 Theemphssis here is on t he m o d u l a r angle since thinis an argument of the Tablea. A ll formulae concerningI a n d e n ' s t ramformat ion mny ab0 be expressed in t e r mof t h e modulus k=m) =sin a and ita complement k '= m != = C o o a.


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598 ELLIPTIC I N T E G R A L 8Thus the step from n to n+ 1 decreases the modularangle but increases the amplitude. By iteratingthe pr ocess we can descend from a given modularangle to one whose magnitude is negligible, when17.4.19 becomes applicable.With %= awe have

17.5.4 F((p\ a)= 2-" 6 (l+ sina,)~((p,,\ a,)S = l

m17.5.5 F((p\ a)= @ I I (l+ sin a,)8-1

1 P nn + m 2" n+ - 2"7.5.6 @= lim - (cp,\ a,,)=lim -

017.5.7 K= F(b\ a)= I I (1 + sin a,)a-117.5.8 F((p\ a)= 2r-'[email protected]

1 1E((p\ al= ~( 'p \ a)I z sin*a (1 z sin a l1+ $ sin a l sin a,+ . . .)]+ sin a[i (sin a l)1 / 2sin (p l

122- (sin a l sin a 2) 1/ 2in e + . . .17.5.10

122in a l+ - sin a l sin a 21+ g sin a l sin a2sin as+ . . .

Amending Landen TraneformationLet a m , ,,+ lbe two modular angles such that

17.5.11 ( l+ sh a ,, )( l+ m~ ,,+ 1)= 2 (a,,+ + l>a,)and let (pn, (P,,+ ~ be two corresponding amplitudessuch that17.5.12 sin (2qm+ l--(p,) = s i n amsin p,, (Pn+ l


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599LLIPTIC INTEGRALS

To determine K ' ( a ) , E ' ( a ) we start with17.6.5whence

d= 1,bi= sin a, C;= COS a

1K' (u ) = -2a;v7.6.617.6.7

To calculate F ( q\ a ), E ( v\ a ) start from 17.5.2which corresponds to the descending Landentransformation and determine +e,, (p1, . . . , vNsuccessively from the relation17.6.8 tan ( v n+ t - vn ) = ( b n / a n ) tan vnpn, &= v

Then to the prescribed accuracy*17.6.9 F(v\ a)= cs y/ ( 2N a N )

a v \ 4 = E(v\ a)- (-/ rnF(v\ a)17.6.10* = cI sin e+ cZ in pi+ . . . + cN s in pN

17.7. Elliptic Integrals of the Third Kind17.7.1n(n;v\ a) =J ' 1 -n sin*e)-l(i -sin2 a sin*e) + te

17.7.2 I I ( n; 4 ?r\ a)= I ( n \ a )Case (i) H y p e r b o l i c CMC O < n < sin' P

0

17.7.4

17.7.5

In the above we can also use Neville's thetafunctions 16.36.17.7.6 lI(n\ a) = K (u ) + & K ( a )Z (r \ a )

CMC (ii) H y p e r b o l i c Gee I>The case n>l can be reduced to the caseO - n ( N \ a )

Cam? (iii) C i r c u l a r CMC sin' P< Ir = a r c a in [(l-n)/ Cos' a]' O l e < 4B= ) ?r F ( e\ \ 90 -a )/ K ( a)q = ! z( d

17.7.10v= 4 rF (v\ a) K(a) &= [n( 1- ) - '(n- inz a)-']*

17.7.11 -n(n; v \ a >= &(A-4/ 417.7.12X=arctan (tanh B tan v)

+ 2 f: -1)'-'8-'p(l-p)-'in 2m sinh 2sB8 -1

17.7.13

17.7.14wh e r e A, is Heurnan'sLa m bda function, 17.4.39.

II(n\ a)= K (a)+ 41&[1 -A~(e\ a)]


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6 0 0 ELLIPTIC

F I GURE7.11. EUipt ic in h gr d of the third k i n dW n; cp\a).Case ( iv) C i r c u l a r CP W n l

17.7.21 a= u/ 2n( n; cp\ r/ 2) = (1 -n)-'[ln (tan q+ sec q)

-3n i In ( l+ n + in cp)( l-n# sin c p ) - ' ] n # 117.7.22 n = fs in a( 1 7 s i n a ) { 2 n ( f s in a; q \ a )-F ( q \ a )}

= arc tan [( lT -sin a) t an q/ A(cp)]17.7.23 n = l fc o s a2 cos d ( 1 f c o s a; q \ a ) = fi I n [( l+ t a n q

*A(cp))(l-tan P - A ( v ) ) - ~I + ~ n [(A(v)+ c o s a- an cp)(A(cp)-cos a t an q)-']T - (1 Fcos a)F(cp\ a)

17.7.24 n= sina aII(sin' a;cp\a)= ' aE(p\ a) -(tanz a sin 2q) (2A( p))17.7.25 n = lI I ( 1 ; c p \ a ) = F ( q \ a ) - s e . c z a E ( c p \ a ) + s e c 2 a t an cpA(cp)

17.8. Us e and Extension of the TablesExample 1. Reduce to canonical form Jy-I&whereV= -3Z'+ 3428--119Z'+ 1722-90

By inspection or by solving an equation of thefour th degree we find t hat~ ' = Q I Q ~Where Q1= 32P-IOZ+ g, Q2= -Z'+ &-IO

First Methodfect ~ - X Q ~ = ( ~ + X ) ~ P - ( ~ O + ~ X ) Z + ~ + ~ O Xquare if the discrim inant b 8 pW -

2 1(10+ 8X)2-4(3+X)(9+ 10h)=0; i.e ., if A= -- or 33and then

2 7 1 7Q1+3Qz=g (z-l)', Q1-3 Q z = z (2-2)'

Solving for Q1 and Qz wegetQ1= (2- 1)'+ 2 (2-2)*, Qz= 2 2 - 1y-3 (2-2)'T h e subst i tut ion t = 2 - I) / @-% then gives

Sy-'dz= *S[(t '+ 2)(2tD-3)1-Uf


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601LLIPTIC I N T E G R A L SIf the quart ic y2= 0 h as four real roots in 2(or in t he case of a cubic all th ree roots are real),we must so combine the factors t hat no root ofQl= O l i e s between the roots of Q l= O and no rootof Ql= O l i e s between the roots of Ql= O. Providedthis condition is observed the method just de-scribed wi l l always lead to real va lues of A. These

values may, however, b e irr at ional.w r i t e

Sccond Method

and let t he discrim inant of Q2t*-Ql be4 T a = 8t2+ 10)*--4(t 2+ 3)(10 t4+ 9)

= 4( 3t 2+ 2)(2tP-1)Then~y - d z = * ~T -L d t = f~[( 3 t 2 + 2 ) ( 2t s -l ) ]- t d lThis method wi l l succeed if, as here, T2 as afunction of t 2 has r eal factors. If the coefficientsof the given quartic are rational numbers, thefactors of T2wi l l likewise be ra t ional.

w r i t e Third Method

an d let th e discrim inan t of Q2w-Q1 be4 W = 4 ( 3 ~+ 2 ) 2~-1)= 4( Ad + B w+ C)

T hen ifzZ= W/ w and 2 2= (B-zZ )2-4 AC = (2 1-l)3+ 4 8

H owever, in t his case th e fact ors of 2 are complexand th e method fails.Of the second and th ird m eth ods onewill alwayssucceed where the ot her fails, and if t he coefficient sof the given quartic are rational numbers, thefactors of T a or 22, a s the case may be, will berational.Example 2. Reduce to canonical form y- &where y2= z(z- 1) (2-2).We use the third method of Example 1 takingQ i = ( ~ -l ) , Q ~ = s ( z - ~ ) a nd w r it in g

T he d iscr imin an t of Q2w- Q1= 2w- (2w+ 1)2+ 1iss o t h a t 4 W = ( 2 ~ + ) 2 -4~= 4+ + 1 1W = A d + B w + C where A= l, B=O, C= 4and if we write zZ= W/w and22= (B--zZ) 4AC= (zZ)2- 1= (z2--1)(zZ+ I ),

J y- dz= *S[(zZ-l)(zZi.1)]- zThe first method of Example 1 fails with theabove values of Ql and Q2 since the root of Ql= Ol i e s between the roots of Qa= O, and we getimaginary values of A. The method succeeds,however, if we t ake Q1= z, Q2= (2-1)(2-2), forthen the roots of Ql= O do n ot lie between t hoseof Q2= O.Example 3. Find K(80/ 81) .

h t ethodUse 17.3.29 with m = 80/ 81, m1= l / 81, m:/ = 1/ 9Since [( l -mi/ ) (1 + m:/ )- I2 .64, K (80/ 81) =1.8 K (.64) = 3.59154 500 to 8D, t aking K( .64) fromTable 17.1.

Sccond MethodTable 17.4 giving L(m) is useful for comp ut ingK(m) when m is near unity or K (m )when m isnear zero.K(80/ 81)= K (80/ 81) (16X81)--L(80/ 81).By interpolation in Tablea 17.1 and 17.4, since?rSol81 = .98765 43210,

K (80/ 81) 1.57567 8423L(80/ 81) = .00311 16543K(80/ 81)= r - ( 1.575678423) (7.16703 7877)

-.00311 16543= 3.59154 5000 to 9D .Third Method

T he polynomial app roximat ion 17.3.34 gives to8DK (80/ 81)= 3.59154 501

Fourth Method, Arithket ic-Geometr ic M ernH ere sin2 a = 80/ 81 and we star t with

1%= l, o=,, c o = W = . 9 9 3 8 0 79900giving


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602 ELLIPTIC INTEORALS

0123

1. oomo 0 0 0 0 0 . l l l l l 11111 .99380 79900. 6 6 5 6 6 6 6 6 6 5 .33333 33333 .44444 44444.44444 44444 I .43033 14829 I . 11111 11111

1.39994 42026.99994 42041.99994 42041

.43738 79636 .43733 10380 .00705 64808.43736 95008 .43736 94999 oooO2 84628.43736 95003 .43736 95003 0'6 I I

123

1T h u s K ( 80/ 81) = 3 ~ ~ ; ' = 3 . 5 9 1 5 4 001.Example 4. Find E (80/ 81) .

Firat MethodUrn 17.3.30 wh ich gives, with m = 80/ 81

E(80/ 81)=; E(.64)-g K(.64)= 1.01910 6047

taking E(.64)and K (.64) from T ab le 17.1.Sccond Method

P o lynom ia l ap p r ox ima t ion , 17.3.36 g ivesE(80/ 81)= 1.01910 6060. T he lest t wo figuresm ust be dropped t ok ee p within the limit ofaccuracy of t he meth od.

0 0.(3)26841 250433)26837 66:!3)26837 66

Third MethodAr ithm et ic-gwm etr ic mean, 17.6. T he numbera

were calculated in Ex am ple 3, fourt h m ethod, andwe haveK(80/ 81) -E(80/ 81) 1= - [4 + 2 4+ 2 *4 + . . . + 2%]K(80/ 81) 2

1= %1.43249 712981= .71624 85649.

Using the value of K(80/ 81) found in Ex am ple 3,fourt h method, we haveE(80/ 81) = 1.01910 6048 to 9D .

Example 5.Here m l= .0005 and s o from T able 11.4Find pwhen m= .9995.&(m) = .06251 563013~1= m ,&(m )= .00003 2578 15.

From 17.3.19h (i)=# /hi)= r' / 10.37324 1132

= .95144 84701q= .38618 125.

The computation could also be made usingcom m on logar ith m s with th e aid of 17.3.20. T hepoint of this procedure is tha t i t enab les us tocalculate q l without the loss of significant figureawhich would result from direct interpolation inTable 17.1. By this means In ( /p l ) can be foundwithout loss of accuracy.Example 6. Find m to 10D when K ' / K= .25and when K'/K= .5 .Fr om 17.3.15 with K '/ K = .25 we can write theiteration formulam'"+"= 1- 6 ~ " xp [ W L(~'" ' ) / K ' (~' " )

Then by iteration using Tables 17.1 and 17.4

T hu s m = .99994 42041.iter ation form ula,From 17.3.16 with K ' / K = 3.5 we can write t he

I

T hus m= .00026 83765.The above methods in conjunction with theauxiliary Table 17.4 of L ( m) enable us to extendT ab le 17.3 for K '/ K >3, and for K'lK


17/21

E L L IPT IC I N T E G R A L S 603

35'40'45'50'

56' 1 58' 1 60'. 6 3 8 0 3 .63945 .64085. 73914 .74138 . 74358.84450 .84788 .85122.95479 .95974 .96465

A--

437509

From this we form the table of F((p\ 56.789089')

35'40'45'50'

9 I F.63859.74003.84584.95674

A

101441058111090

a,

7 2

A rough estimate now shows that q lioa between40' and 41'. We therefore form the followingtable of F((p\ 56.789089') by direct interpolationin the foregoing table9 F40.0' .7400340.5' .7504041.0' .76082

whence by linear inverse interpolation.75.342- .75040 = 40.64490.76082- .750401p= 40.5'+ .5' [and s o sin (p= .65137= sn (.753421.7).This method of bivariate interpolation is givenmerely as an illustrat ion. Other more directmethods such as that of the arithmetic-geometricmean described in 17.6 and illustrated for theJacobian functions in chapter 16 are less laborious.Example 8. Evaluate

F h t Method, Bivar iate In terpolat ionFrom 17.4.50 we have&s' ( 2 t 2 +l ) ( t 2 - 2 ) ] - " 2 d t =F ( ~ \ a ) - F ( $ 3 \ U,

where1 J z 4inza= -' cos ql= -, cos e = -5 3 2

Thus a= 26.56505 12') (pl= 61.87449 43') ~ = 4 5 ' ,F(*\ a)= 1.115921 and F(cpl\ a)= .800380 andtherefore the integral is equal to .141114.

Second M ethod, Nu m erical Quadrat ureSimpson'sformula with 11 ordinates and intervalExample 9. Evaluate.1 gives .141117.

J [(tz-2)(t2-4)]-'dt.2

F irst Method, R eduction t o Standard F o r m andBivariate InterpolationHere we can use 17.4.48 noting that az-4,b2= 2, and t hat

s o' t2-2) (t2--4)]-'dt=12- [1.854075-.535623]= .659226

2 . 2 . 2sin ( p 1 = - 1 ~1 1 1 e = - , in2a= --.2 4 4a= 45', a= 90 ,~= 3 0' .

Seeond Method, N um erical Int egrat ion

whereThus

If we wish to usenumerical integrat ion we mustobserve that the integrand has a singularity att = 2 where it behaves like [8(t-2)]-'.We remove the singularit y at t = 2, by writingL4 z-2) ( 2- 4) ]%!t = l f( t )dt+ J 48( -2)]-Utwhere

Z

f )= [ *- 2)( ' - 4) - - 8( - ) -.If we definef (2)=O,

can be calculated by numerical quadrature. AlsoI48 t- )]-Mt =[ $ t- ) I= 1and thus we calculate the integral as

1+ f t ) d t= 1- 340773= .659227.14Example 10. Evaluate

2 - 7 ~+ 6 = ( ~ - 1) ( ~ -2 ) ( ~ + 3 ) nd we u s e 17.4.65with &= 2, &= 1, p*= -3,


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6 0 4 ELLIPTIC I N T E G R A L S

123

m= sin2 u= 4/ 5, X4%/ 2, os' ~= 3 / 4 .Thusa= 63.434949' , c0=3O0 and

~= 2(5)-+ F (30' \ 63.434949' )= 2(5) -+ ( .543604)= .486214from Table 17.5.The above integral is of the Weierstraas type andExample 11. Evaluatein fact 17= @( h ; 8, -24) (see chapter 18).

.33333 333 .94280 904,02943 725 .99956 663.O0021 673 .99999 998

s,"" (24-12t+ 2t2-t3)-1'P d t .We have

24- 2t+ 2t2- 8= - t- )(P+ 12)= - (t )There is only one real zero and we thereforeuse 17.4.74 with P(t)= tS-2t2+12t-24, @= 2 so

that P'(2) = 16, P" (2) = 8, X=2 and therefore1m= sin2 CY-ZI a= 30 .

Therefore the given integrr tl'is

where*= 70.52877 93'a= 3'

1c o s e = - j e = 60and the integral= i[1.510344- 1.2125971= 148874.Example 12. Use Landen's tr ansformat ion toevaluate

L ' '1 -: in28)-1'2 dB to 5D.First Method, Demcending Tramformation

We me 17.5.1 to give

1+ cos 30'- 1.0717971+ sin a l=COS C Y ~ = [( ~ - S~IJ (l+ sh ~~l)]" ~= .997419

= 1.001292;COS ~~2= .9999991+ sin a2= 1+ cos CY11+ sin CY3 = 1.0000001+ cos a2Thus from 17.5.7,

the integral=F(9O0\ 30') = z (1.071797) (1.001292)2= 1.68575 to 5D.

Second Method, Aacendiqg T d o n u a t i o nWe use 17.5.11 to give

I+ C O S CYn+l=2/ (1+& CY

sin ( 2a- 0') = sin 30 , n= 60'sin (2cp,-~1)= sba1sin a, e= 57.367805'sin ( 2 e - e) sin sin e, *= 57.348426'sin (2cp4-~)= sin a s sin e ,From 17.5.16

v4= 57.348425'= @.

2 2 2F(90a \ 300)= E.94280 904 1.99956 663In tan(,,+;@)1.99999 998

= 1.37288050 I n tan 73.674213'= 1.37288 050(1.22789 30)

F(9Oo\ 3O0)= 1.68575 to 5D.Example 13. Find the value of F(89.5'\ 89.5').First Method

This is a case where interpolation in Table 17.5is not possible. We use 17.4.13 which givesF (89.5'\ 89.5')= F(90'\ 89.5') -F(# \ 89.5')

where cot *= sin (3') cot (.5O)= cos (5)$= 45.00109 084'

and P'(*\ 89.5') = 881390 from Table 17.5.F(90'\ 89.5') = K(sh' 89.5') = K(.99992 38476)

= 6.12777 88Thus F(89.5'\ 89.5') = 5.246389.

Seeond MethodLanden's ascending transformation, 17.5.11,gives


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ELLIPTIC INTEGRALS 6 0 5COS al= (l-sin 89.5')/ (l+ sin 89.5')sin q= [ 1 -COS a l) 1 + C O S al)]*= .99999 99997COS a 2= 0sin q = l .

17.5.12 'then givessin (2fi-89.5')= sh 89.5' sin 89.5'

= 99992 384762fi-89.5'= 89.2929049', fic89.39645 245'

sin @(pn-fi)= sin a1 sin fi, (pn= 89.39645 602'sin (2cpa-(pn)= sin (pn, (Pa= yL1=@T h u s 17.5.16 givesF(89.5'\ 89.5') =

> n (tan 89.69822 801')= 5.24640.($9996 19231Example 14. Evaluate

From 17.4.51 the given integral

where sin a= # ) a= 36.869900Si n &, fi= 48.18968'

By bivariate interpolation in Table 17.6 wefind that the given integral= ;[ . 8 0 9 0 4- 41192]= .0O496.Simpson'srule-with 3 ordinates gives

6 [005O4+ 01 975+ OO5]= 00496.Example 15. EvaluateII (A; 45'\ 30) =

( l -&~in~O)-~(l - -+in00)-*d8 o 6D.

This is case (i) of int egrals of the third kind,O


20/21

ELLIPTIC INTEGRALS0 6c= armin [(l-n)/ cosa a]$= 45O@= 4~F( 45~\ 60~) / K(30~)79317 74~= ~~F ( 4 5" \ 3 0~) / r ( ( 3 0~)74951 51

&= (40/ 9)*q= .O1797 24

and 80 from 17.7.1111 (# ; 45'\ 3O0)= (40/ 9)1'2(h-4~)

~2 . 10818 1 .55248 32-4(.03854 26)(.74951 51) 1 = .921129.

Table 17.9 gives .92113 with 4 point Lagrangianinterpolation.Example 18. Evaluate the complete elliptici n t q p l I I (# \ 30) o 5D.F'rom 17.7.14 we have

I I (# \ 30)= K(30)+ E 8 l A0(e\ 3Oo) Jwhere c= arcsin [ l -n ) / co~~a]~/ ~= 45~.hus usingTable 17.8

I I (# \ 30 )= 2.80099.Table 17.9 gives 2.80126 by 6 point Lagrangianinter polation. The discrepancy results from in -

terpolation with respect to n for p= 90 in Table17.9.Example 19. EvaluateI I ( i ) ; 450\ 30)t o 5D.

= I " (1 sins e) 1(1-+ sin2e) 112 d~

Heren = z (p=45O,a= 30 and since the character-istic is greater t han unity weuse 17.7.7

N= n-I sin2a= .2, pl= (1/ 5)*I I ( i ) ; 45O\3Oo) - I I ( 2; 45O\ 3Oo) + F(45'\ 3O0)

= - .83612+.8O437

= 1.13214.Numerical quadrature gives the same result.Example 20. Evaluate

Here the characteristic is negat ive aHd we there-1 1fore use 17.7.15 with n= -- sinaa= -4' 4

Using Tables 4.14, 17.5, and 17.9 we getI I (-*; 45O\ 3Oo)= 76987

ReferencesTCSt8

[17.1] A. Csyley, An elementary treatise on ellipticfunctions (Dover Publications, Inc., New York,N.Y., 1956).(17.21 A. Erdblyi et al., Higher t ranscendental functions,vol. 2, oh. 23 (McGraw-Hill Book Co., Inc.,New York, N.Y., 1953).(17.31 L. V. King, On the direct numerical calculation ofelliptic functions and integrals (CambridgeUniv. Press, Cambridge, England, 1924).l17.41 E. H. Neville, Jacobian elliptic functions, 2d ed.(Oxford Univ. Press, L o n d o n , England, 1951).(17.5) F. Oberhet t inger and W . Magnus, Anwendung derelliptisohen Funktionen in Physik und Technik(Springer-Verlsg, Berlin, Germany, 1949).

[17.6] F. Tricomi, Elliptische Funktionen (AkademiacheVerlagsgesellschaft , Leipzig, Germany, 1948).[17.7] E. T. Whittaker and G. N. Watson, A course ofmodern analysis, chs. 20, 21, 22, 4th ed. (Cam-bridge Univ. Press, Cambridge, England, 1952).

Tables117.8) P . F . Byrd and M. D. Friedman, Handbook ofelliptic integrals for engineers and physicists(Springer-Verlag, Berlin, Germany, 1954).[17.9] C. Heuman, Tables of complete elliptic integrals,

(17.101 J. Houel, Recueil de formules et de t ables numb-J. Math. Phys. 20, 127-206 (1941).riques (Gauthier-Villars, Paris, France, 1901).


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E L L I P T I C INTEGRALS 6 0 7[17.11] E. Jahnke and F. Em&, Tables of functions, 4thed. (Dover Publications, Inc., New York, N.Y.,1945).[17.12] L. M. Milne-Thomson, Jacobian elliptic functiontables (Dover Publications, Inc., New York,N.Y., 1956).(17.131 L. M. Milne-Thomeon, Ten-figure table of thecomplete elliptic integrals K, K ,E, E nd a

9 Roc. h n d o n M a t h .table Of 8m d mS oc . , 33 (1931).

(17.141 L. M. Milne-Thornson, T h e Zeta function ofJacobi, Roc. Roy. S o c . Edinburgh 52 (1931).[17.15] L. M. Milne-Thomson, Di e elliptischen Funktionenvon Jacobi (Julius Springer, Berlin, Germany1931).[17.16] K. Pearson, Tabla of the complete and incompleteellipt ic integrals (Cambridge Univ. Press, Cam-bridge, England, 1934).[17.17] G. W . and R. M. Spenceley, Sm ith sonian ellipticfunction tablea, Smithsonian Miscellaneous Collection, vol. 109 (Washington, D.C., 1947).

thompson. elliptical integrals

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