TOPICS
Search

Elliptic Integral of the First Kind

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

Let the elliptic modulus k satisfy 0<k^2<1, and the Jacobi amplitude be given by phi=amu with -pi/2<phi<pi/2. The incomplete elliptic integral of the first kind is then defined as

u=F(phi,k)=int_0^phi(dtheta)/(sqrt(1-k^2sin^2theta)).
(1)

The elliptic integral of the first kind is implemented in the Wolfram Language as EllipticF[phi, m] (note the use of the parameter m=k^2 instead of the modulus k).

Letting

t=sintheta
(2)
dt=costhetadtheta
(3)
=sqrt(1-t^2)dtheta,
(4)

Equation (1) can be written as

F(phi,k)=int_0^(sinphi)1/(sqrt(1-k^2t^2))(dt)/(sqrt(1-t^2))
(5)
=int_0^(sinphi)(dt)/(sqrt((1-k^2t^2)(1-t^2))).
(6)

Letting

v=tantheta
(7)
dv=sec^2thetadtheta=(1+v^2)dtheta,
(8)

then the integral can also be written as

F(phi,k)=int_0^(tanphi)(dv)/(sqrt((1+v^2)(1+k^('2)v^2))),
(9)

where k^('2)=1-k^2 is the complementary elliptic modulus.

The inverse function of F(phi,k) is given by the Jacobi amplitude

F^(-1)(u,k)=phi=am(u,k).
(10)

The integral

I=1/(sqrt(2))int_0^(theta_0)(dtheta)/(sqrt(costheta-costheta_0)),
(11)

which arises in computing the period of a pendulum, is also an elliptic integral of the first kind. Use

costheta=1-2sin^2(1/2theta)
(12)
sin(1/2theta)=sqrt((1-costheta)/2)
(13)

to write

sqrt(costheta-costheta_0)=sqrt(1-2sin^2(1/2theta)-costheta_0)
(14)
=sqrt(1-costheta_0)sqrt(1-2/(1-costheta_0)sin^2(1/2theta))
(15)
=sqrt(2)sin(1/2theta_0)sqrt(1-csc^2(1/2theta_0)sin^2(1/2theta)),
(16)

so

I=1/2int_0^(theta_0)(dtheta)/(sin(1/2theta_0)sqrt(1-csc^2(1/2theta_0)sin^2(1/2theta))).
(17)

Now let

sin(1/2theta)=sin(1/2theta_0)sinphi,
(18)

so the angle theta is transformed to

phi=sin^(-1)[(sin(1/2theta))/(sin(1/2theta_0))],
(19)

which ranges from 0 to pi/2 as theta varies from 0 to theta_0. Taking the differential gives

1/2cos(1/2theta)dtheta=sin(1/2theta_0)cosphidphi,
(20)

or

1/2sqrt(1-sin^2(1/2theta_0)sin^2phi)dtheta=sin(1/2theta_0)cosphidphi.
(21)

Plugging this in gives

I=int_0^(pi/2)1/(sqrt(1-sin^2(1/2theta_0)sin^2phi))(sin(1/2theta_0)cosphidphi)/(sin(1/2theta_0)sqrt(1-sin^2phi))
(22)
=int_0^(pi/2)(dphi)/(sqrt(1-sin^2(1/2theta_0)sin^2phi))
(23)
=K(sin(1/2theta_0)),
(24)

so

I=1/(sqrt(2))int_0^(theta_0)(dtheta)/(sqrt(costheta-costheta_0))
(25)
=K(sin(1/2theta_0)).
(26)

Making the slightly different substitution phi=theta/2, so dtheta=2dphi leads to an equivalent, but more complicated expression involving an incomplete elliptic integral of the first kind,

I=21/(sqrt(2))1/(sqrt(2))csc(1/2theta_0)int_0^(theta_0)(dphi)/(sqrt(1-csc^2(1/2theta_0)sin^2phi))
(27)
=csc(1/2theta_0)F(1/2theta_0,csc(1/2theta_0)).
(28)
EllipticFReIm
EllipticFContours

Therefore, the identity

F(z,cscz)=sinzK(sinz)
(29)

holds over at least some region of the complex plane. The region of applicability is -pi/2<R[z]<pi/2, which is shown above.

The elliptic integral of the first kind satisfies

F(-phi,k)=-F(phi,k).
(30)

Special values of F(phi,k) include

F(0,k)=0
(31)
F(1/2pi,k)=K(k),
(32)

where K(k) is known as the complete elliptic integral of the first kind.

See also

Complete Elliptic Integral of the First Kind, Elliptic Characteristic, Elliptic Integral of the Second Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value, Elliptic Modulus, Gauss's Transformation, Jacobi Amplitude, Landen's Transformation, Legendre Relation, Modular Angle, Parameter

Related Wolfram sites

http://functions.wolfram.com/EllipticIntegrals/EllipticF/

Explore with Wolfram|Alpha

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.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Spanier, J. and Oldham, K. B. "The Complete Elliptic Integrals K(p) and E(p)" and "The Incomplete Elliptic Integrals F(p;phi) and E(p;phi)." Chs. 61-62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987.Tölke, F. "Parameterfunktionen." Ch. 3 in Praktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83-115, 1966.Tölke, F. "Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gattung. Elliptische Amplitudenfunktionen sowie Legendresche F- und E-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktionen," and "Normalintegrale dritter Gattung. Legendresche Pi-Funktion. Zurückführung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Chs. 6-7 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 58-144, 1967.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.

Referenced on Wolfram|Alpha

Elliptic Integral of the First Kind

Cite this as:

Weisstein, Eric W. "Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html

Subject classifications