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Elliptic Integral


An elliptic integral is an integral of the form

 int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx,
(1)

or

 int(A(x)dx)/(B(x)sqrt(S(x))),
(2)

where A(x), B(x), C(x), and D(x) are polynomials in x, and S(x) is a polynomial of degree 3 or 4. Stated more simply, an elliptic integral is an integral of the form

 intR(w,x)dx,
(3)

where R(w,x) is a rational function of x and w, w^2 is a function of x that is cubic or quartic in x, R(w,x) contains at least one odd power of w, and w^2 has no repeated factors (Abramowitz and Stegun 1972, p. 589).

Elliptic integrals can be viewed as generalizations of the inverse trigonometric functions and provide solutions to a wider class of problems. For instance, while the arc length of a circle is given as a simple function of the parameter, computing the arc length of an ellipse requires an elliptic integral. Similarly, the position of a pendulum is given by a trigonometric function as a function of time for small angle oscillations, but the full solution for arbitrarily large displacements requires the use of elliptic integrals. Many other problems in electromagnetism and gravitation are solved by elliptic integrals.

A very useful class of functions known as elliptic functions is obtained by inverting elliptic integrals to obtain generalizations of the trigonometric functions. Elliptic functions (among which the Jacobi elliptic functions and Weierstrass elliptic function are the two most common forms) provide a powerful tool for analyzing many deep problems in number theory, as well as other areas of mathematics.

All elliptic integrals can be written in terms of three "standard" types. To see this, write

R(w,x)=(P(w,x))/(Q(w,x))
(4)
=(wP(w,x)Q(-w,x))/(wQ(w,x)Q(-w,x)).
(5)

But since w^2=f(x),

Q(w,x)Q(-w,x)=Q_1(w,x)
(6)
=Q_1(-w,x),
(7)

then

wP(w,x)Q(-w,x)=A+Bx+Cw+Dx^2+Ewx+Fw^2+Gw^2x+Hw^3x
(8)
=(A+Bx+Dx^2+Fw^2+Gw^2x)+w(c+Ex+Hw^2x+...)
(9)
=P_1(x)+wP_2(x),
(10)

so

R(w,x)=(P_1(x)+wP_2(x))/(wQ_1(w))
(11)
=(R_1(x))/w+R_2(x).
(12)

But any function intR_2(x)dx can be evaluated in terms of elementary functions, so the only portion that need be considered is

 int(R_1(x))/wdx.
(13)

Now, any quartic can be expressed as S_1S_2 where

S_1=a_1x^2+2b_1x+c_1
(14)
S_2=a_2x^2+2b_2x+c_2.
(15)

The coefficients here are real, since pairs of complex roots are complex conjugates

[x-(R+Ii)][x-(R-Ii)]=x^2+x(-R+Ii-R-Ii)+(R^2-I^2i)
(16)
=x^2-2Rx+(R^2+I^2).
(17)

If all four roots are real, they must be arranged so as not to interleave (Whittaker and Watson 1990, p. 514). Now define a quantity lambda such that S_1-lambdaS_2

 (a_1-lambdaa_2)x^2-(2b_1-2b_2lambda)x+(c_1-lambdac_2)
(18)

is a square number and

 2sqrt((a_1-lambdaa_2)(c_1-lambdac_2))=2(b_1-b_2lambda)
(19)
 (a_1-lambdaa_2)(c_1-lambdac_2)-(b_1-lambdab_2)^2=0.
(20)

Call the roots of this equation lambda_1 and lambda_2, then

S_1-lambda_2S_2=[sqrt((a_1-lambda_2a_2)x^2)+sqrt(c_1-lambda_2c_2)]^2
(21)
=(a_1-lambda_2a_2)(x+sqrt((c_1-lambda_2c_2)/(a_1-lambda_2a_2)))
(22)
=(a_1-lambda_2a_2)(x-beta)^2
(23)
S_1-lambda_1S_2=[sqrt((a_1-lambda_1a_2)x^2)+sqrt(c_1-lambda_1c_2)]^2
(24)
=(a_1-lambda_1a_2)(x+sqrt((c_1-lambda_1c_2)/(a_1-lambda_1a_2)))
(25)
=(a_1-lambda_1a_2)(x-alpha)^2.
(26)

Taking (25)-(26) and lambda_2(1)-lambda_1(2) gives

S_2(lambda_2-lambda_1)=(a_1-lambda_1a_2)(x-alpha)^2-(a_1-lambda_2a_2)(x-beta)^2
(27)
S_1(lambda_2-lambda_1)=lambda_2(a_1-lambda_1a_2)(x-alpha)^2-lambda_1(a_1-lambda_2a_2)(x-beta)^2.
(28)

Solving gives

S_1=(a_1-lambda_1a_2)/(lambda_2-lambda_1)(x-alpha)^2-(a_1-lambda_2a_2)/(lambda_2-lambda_1)(x-beta)^2
(29)
=A_1(x-alpha)^2+B_1(x-beta)^2
(30)
S_2=(lambda_2(a_1-lambda_1a_2))/(lambda_2-lambda_1)(x-alpha)^2-(lambda_1(a_1-lambda_2a_2))/(lambda_2-lambda_1)(x-beta)^2
(31)
=A_2(x-alpha)^2+B_2(x-beta)^2,
(32)

so we have

 w^2=S_1S_2=[A_1(x-alpha)^2+B_1(x-beta)^2][A^2(x-alpha)^2+B^2(x-beta)^2].
(33)

Now let

t=(x-alpha)/(x-beta)
(34)
dt=[(x-beta)^(-1)-(x-alpha)(x-beta)^(-2)]dx
(35)
=((x-beta)-(x-alpha))/((x-beta)^2)dx
(36)
=(alpha-beta)/((x-beta)^2)dx,
(37)

so

w^2=(x-beta)^4[A_1((x-alpha)/(x-beta))^2+B_1][A_2((x-alpha)/(x-beta))+B_2]
(38)
=(x-beta)^4(A_1t^2+B_1)(A_2t^2+B_2),
(39)

and

w=(x-beta)^2sqrt((A_1t^2+B_1)(A_2t^2+B_2))
(40)
(dx)/w=[((x-beta)^2)/(alpha-beta)dt]1/((x-beta)^2sqrt((A_1t^2+B_1)(A_2t^2+B_2)))
(41)
=(dt)/((alpha-beta)sqrt((A_1t^2+B_1)(A_2t^2+B_2))).
(42)

Now let

 R_3(t)=(R_1(x))/(alpha-beta),
(43)

so

 int(R_1(x)dx)/w=int(R_3(t)dt)/(sqrt((A_1t^2+B_1)(A_2t^2+B_2))).
(44)

Rewriting the even and odd parts

R_3(t)+R_3(-t)=2R_4(t^2)
(45)
R_3(t)-R_3(-t)=2tR_5(t^2),
(46)

gives

R_3(t)=1/2(R_(even)-R_(odd))
(47)
=R_4(t^2)+tR_5(t^2),
(48)

so we have

 int(R_1(x)dx)/w=int(R_4(t^2)dt)/(sqrt((A_1t^2+B_1)(A_2t^2+B_2)))+int(R_5(t^2)tdt)/(sqrt((A_1t^2+B_1)(A_2t^2+B_2))).
(49)

Letting

u=t^2
(50)
du=2tdt
(51)

reduces the second integral to

 1/2int(R_5(u)du)/(sqrt((A_1u+B_1)(A_2u+B_2))),
(52)

which can be evaluated using elementary functions. The first integral can then be reduced by integration by parts to one of the three Legendre elliptic integrals (also called Legendre-Jacobi elliptic integrals), known as incomplete elliptic integrals of the first, second, and third kinds, denoted F(phi,k), E(phi,k), and Pi(n;phi,k), respectively (von Kármán and Biot 1940, Whittaker and Watson 1990, p. 515). If phi=pi/2, then the integrals are called complete elliptic integrals and are denoted K(k), E(k), Pi(n;k).

Incomplete elliptic integrals are denoted using a elliptic modulus k, parameter m=k^2, or modular angle alpha=sin^(-1)k. An elliptic integral is written I(phi|m) when the parameter is used, I(phi,k) when the elliptic modulus is used, and I(phi\alpha) when the modular angle is used. Complete elliptic integrals are defined when phi=pi/2 and can be expressed using the expansion

 (1-k^2sin^2theta)^(-1/2)=sum_(n=0)^infty((2n-1)!!)/((2n)!!)k^(2n)sin^(2n)theta.
(53)

An elliptic integral in standard form

 int_a^x(dx)/(sqrt(f(x))),
(54)

where

 f(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,
(55)

can be computed analytically (Whittaker and Watson 1990, p. 453) in terms of the Weierstrass elliptic function with invariants

g_2=a_0a_4-4a_1a_3+3a_2^2
(56)
g_3=a_0a_2a_4-2a_1a_2a_3-a_4a_1^2-a_3^2a_0.
(57)

If a=x_0 is a root of f(x)=0, then the solution is

 x=x_0+1/4f^'(x_0)[P(z;g_2,g_3)-1/(24)f^('')(x_0)]^(-1).
(58)

For an arbitrary lower bound,

 x=a+(sqrt(f(a))P^'(z)+1/2f^'(a)[P(z)-1/(24)f^('')(a)]+1/(24)f(a)f^(''')(a))/(2[P(z)-1/(24)f^('')(a)]^2-1/(48)f(a)f^((iv))(a)),
(59)

where P(z)=P(z;g_2,g_3) is a Weierstrass elliptic function (Whittaker and Watson 1990, p. 454).

A generalized elliptic integral can be defined by the function

T(a,b)=2/piint_0^(pi/2)(dtheta)/(sqrt(a^2cos^2theta+b^2sin^2theta))
(60)
=2/piint_0^(pi/2)(dtheta)/(costhetasqrt(a^2+b^2tan^2theta))
(61)

(Borwein and Borwein 1987). Now let

t=btantheta
(62)
dt=bsec^2thetadtheta.
(63)

But

 sectheta=sqrt(1+tan^2theta),
(64)

so

dt=b/(costheta)secthetadtheta
(65)
=b/(costheta)sqrt(1+tan^2theta)dtheta
(66)
=b/(costheta)sqrt(1+(t/b)^2)dtheta
(67)
=(dtheta)/(costheta)sqrt(b^2+t^2),
(68)

and

 (dtheta)/(costheta)=(dt)/(sqrt(b^2+t^2)),
(69)

and the equation becomes

T(a,b)=2/piint_0^infty(dt)/(sqrt((a^2+t^2)(b^2+t^2)))
(70)
=1/piint_(-infty)^infty(dt)/(sqrt((a^2+t^2)(b^2+t^2))).
(71)

Now we make the further substitution u=1/2(t-ab/t). The differential becomes

 du=1/2(1+ab/t^2)dt,
(72)

but 2u=t-ab/t, so

 2u/t=1-ab/t^2
(73)
 ab/t^2=1-2u/t
(74)

and

 1+ab/t^2=2-2u/t=2(1-u/t).
(75)

However, the left side is always positive, so

 1+ab/t^2=2-2u/t=2|1-u/t|
(76)

and the differential is

 dt=(du)/(|1-u/t|).
(77)

We need to take some care with the limits of integration. Write (◇) as

 int_(-infty)^inftyf(t)dt=int_(-infty)^(0^-)f(t)dt+int_(0^+)^inftyf(t)dt.
(78)

Now change the limits to those appropriate for the u integration

 int_(-infty)^inftyg(u)du+int_(-infty)^inftyg(u)du=2int_(-infty)^inftyg(u)du,
(79)

so we have picked up a factor of 2 which must be included. Using this fact and plugging (◇) in (◇) therefore gives

 T(a,b)=2/piint_(-infty)^infty(du)/(|1-u/t|sqrt(a^2b^2+(a^2+b^2)t^2+t^4)).
(80)

Now note that

u^2=(t^4-2abt^2+a^2b^2)/(4t^2)
(81)
4u^2t^2=t^4-2abt^2+a^2b^2
(82)
a^2b^2+t^4=4u^2t^2+2abt^2.
(83)

Plug (◇) into (◇) to obtain

T(a,b)=2/piint_(-infty)^infty(du)/(|1-u/t|sqrt(4u^2t^2+2abt^2+(a^2+b^2)t^2))
(84)
=2/piint_(-infty)^infty(du)/(|t-u|sqrt(4u^2+(a+b)^2)).
(85)

But

2ut=t^2-ab
(86)
t^2-2ut-ab=0
(87)
t=1/2(2u+/-sqrt(4u^2+4ab))
(88)
=u+/-sqrt(u^2+ab),
(89)

so

 t-u=+/-sqrt(u^2+ab),
(90)

and (◇) becomes

T(a,b)=2/piint_(-infty)^infty(du)/(sqrt([4u^2+(a+b)^2](u^2+ab)))
(91)
=1/piint_(-infty)^infty(du)/(sqrt([u^2+((a+b)/2)^2](u^2+ab))).
(92)

We have therefore demonstrated that

 T(a,b)=T(1/2(a+b),sqrt(ab)).
(93)

We can thus iterate

a_(i+1)=1/2(a_i+b_i)
(94)
b_(i+1)=sqrt(a_ib_i),
(95)

as many times as we wish, without changing the value of the integral. But this iteration is the same as and therefore converges to the arithmetic-geometric mean, so the iteration terminates at a_i=b_i=M(a_0,b_0), and we have

T(a_0,b_0)=T(M(a_0,b_0),M(a_0,b_0))
(96)
=1/piint_(-infty)^infty(dt)/(M^2(a_0,b_0)+t^2)
(97)
=1/(piM(a_0,b_0))[tan^(-1)(t/(M(a_0,b_0)))]_(-infty)^infty
(98)
=1/(piM(a_0,b_0))[pi/2-(-pi/2)]
(99)
=1/(M(a_0,b_0)).
(100)

Complete elliptic integrals arise in finding the arc length of an ellipse and the period of a pendulum. They also arise in a natural way from the theory of theta functions. Complete elliptic integrals can be computed using a procedure involving the arithmetic-geometric mean. Note that

T(a,b)=2/piint_0^(pi/2)(dtheta)/(sqrt(a^2cos^2theta+b^2sin^2theta))
(101)
=2/piint_0^(pi/2)(dtheta)/(asqrt(cos^2theta+(b/a)^2sin^2theta))
(102)
=2/(api)int_0^(pi/2)(dtheta)/(sqrt(1-(1-(b^2)/(a^2))sin^2theta)).
(103)

So we have

T(a,b)=2/(api)K(sqrt(1-(b^2)/(a^2)))
(104)
=1/(M(a,b)),
(105)

where K(k) is the complete elliptic integral of the first kind. We are free to let a=a_0=1 and b=b_0=k^', so

 2/piK(sqrt(1-k^('2)))=2/piK(k)=1/(M(1,k^')),
(106)

since k=sqrt(1-k^('2)), so

 K(k)=pi/(2M(1,k^')).
(107)

But the arithmetic-geometric mean is defined by

a_i=1/2(a_(i-1)+b_(i-1))
(108)
b_i=sqrt(a_(i-1)b_(i-1))
(109)
c_i={1/2(a_(i-1)-b_(i-1)) i>0; sqrt(a_0^2-b_0^2) i=0,
(110)

where

 c_(n-1)=1/2a_n-b_n=(c_n^2)/(4a_(n+1))<=(c_n^2)/(4M(a_0,b_0)),
(111)

so we have

 K(k)=pi/(2a_N),
(112)

where a_N is the value to which a_n converges. Similarly, taking instead a_0^'=1 and b_0^'=k gives

 K^'(k)=pi/(2a_N^').
(113)

Borwein and Borwein (1987) also show that defining

U(a,b)=pi/2int_0^(pi/2)sqrt(a^2cos^2theta+b^2sin^2theta)dtheta
(114)
=aE^'(b/a)
(115)

leads to

 2U(a_(n+1),b_(n+1))-U(a_n,b_n)=a_nb_nT(a_n,b_n),
(116)

so

 (K(k)-E(k))/(K(k))=1/2(c_0^2+2c_1^2+2^2c_2^2+...+2^nc_n^2)
(117)

for a_0=1 and b_0=k^', and

 (K^'(k)-E^'(k))/(K^'(k))=1/2(c_0^'^2+2c_1^'^2+2^2c_2^'^2+...+2^nc_n^'^2).
(118)

The elliptic integrals satisfy a large number of identities. The complementary functions and moduli are defined by

 K^'(k)=K(sqrt(1-k^2))=K(k^').
(119)

Use the identity of generalized elliptic integrals

 T(a,b)=T(1/2(a+b),sqrt(ab))
(120)

to write

1/aK(sqrt(1-(b^2)/(a^2)))=2/(a+b)K(sqrt(1-(4ab)/((a+b)^2)))
(121)
=2/(a+b)K(sqrt((a^2+b^2-2ab)/((a+b)^2)))
(122)
=2/(a+b)K((a-b)/(a+b))
(123)
 K(sqrt(1-(b^2)/(a^2)))=2/(1+b/a)K((1-b/a)/(1+b/a)).
(124)

Define

 k^'=b/a,
(125)

and use

 k=sqrt(1-k^('2)),
(126)

so

 K(k)=2/(1+k^')K((1-k^')/(1+k^')).
(127)

Now letting l=(1-k^')/(1+k^') gives

 l(1+k^')=1-k^'=>k^'(l+1)=1-l
(128)
 k^'=(1-l)/(1+l)
(129)
k=sqrt(1-k^('2))
(130)
=sqrt(1-((1-l)/(1+l))^2)
(131)
=sqrt(((1+l)^2-(1-l)^2)/((1+l)^2))
(132)
=(2sqrt(l))/(1+l),
(133)

and

1/2(1+k^')=1/2(1+(1-l)/(1+l))
(134)
=1/2[((1+l)+(1-l))/(1+l)]
(135)
=1/(1+l).
(136)

Writing k instead of l,

 K(k)=1/(k+1)K((2sqrt(k))/(1+k)).
(137)

Similarly, from Borwein and Borwein (1987),

 E(k)=(1+k)/2E((2sqrt(k))/(1+k))+(k^('2))/2K(k)
(138)
 E(k)=(1+k^')E((1-k^')/(1+k^'))-k^'K(k).
(139)

Expressions in terms of the complementary function can be derived from interchanging the moduli and their complements in (◇), (◇), (◇), and (◇).

K^'(k)=K(k^')
(140)
=2/(1+k)K((1-k)/(1+k))
(141)
=2/(1+k)K^'(sqrt(1-((1-k)/(1+k))^2))
(142)
=2/(1+k)K^'((2sqrt(k))/(1+k))
(143)
=1/(1+k^')K((2sqrt(k^'))/(1+k^'))
(144)
=1/(1+k^')K^'((1-k^')/(1+k^')),
(145)

and

 E^'(k)=(1+k)E^'((2sqrt(k))/(1+k))-kK^'(k)
(146)
 E^'(k)=((1+k^')/2)E^'((1-k^')/(1+k^'))+(k^2)/2K^'(k).
(147)

Taking the ratios

 (K^'(k))/(K(k))=2(K^'((2sqrt(k))/(1+k)))/(K((2sqrt(k))/(1+k)))=1/2(K^'((1-k^')/(1+k^')))/(K((1-k^')/(1+k^')))
(148)

gives the modular equation of degree 2. It is also true that

 K(x)=4/((1+sqrt(x^'))^2)K([(1-RadicalBox[{1, -, {x, ^, 4}}, 4])/(1+RadicalBox[{1, -, {x, ^, 4}}, 4])]^2).
(149)

See also

Abelian Integral, Carlson Elliptic Integrals, Complete Elliptic Integral of the First Kind, Complete Elliptic Integral of the Second Kind, Complete Elliptic Integral of the Third Kind, Delta Amplitude, Elliptic Argument, Elliptic Characteristic, Elliptic Function, Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value, Elliptic Modulus, Heuman Lambda Function, Jacobi Amplitude, Jacobi Elliptic Functions, Jacobi Zeta Function, Modular Angle, Nome, Parameter, Weierstrass Elliptic Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Elliptic Integrals." Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.Arfken, G. "Elliptic Integrals." §5.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 321-327, 1985.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Hancock, H. Elliptic Integrals. New York: Wiley, 1917.Kármán, T. von and Biot, M. A. Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems. New York: McGraw-Hill, p. 121, 1940.King, L. V. The Direct Numerical Calculation of Elliptic Functions and Integrals. London: Cambridge University Press, 1924.Prasolov, V. and Solovyev, Y. Elliptic Functions and Elliptic Integrals. Providence, RI: Amer. Math. Soc., 1997.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Elliptic Integrals and Jacobi Elliptic Functions." §6.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 254-263, 1992.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon & Breach, 1986.Timofeev, A. F. Integration of Functions. Moscow and Leningrad: GTTI, 1948.Weisstein, E. W. "Books about Elliptic Integrals." http://www.ericweisstein.com/encyclopedias/books/EllipticIntegrals.html.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Woods, F. S. "Elliptic Integrals." Ch. 16 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 365-386, 1926.

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Elliptic Integral

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Weisstein, Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticIntegral.html

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