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Elliptic Integrals

 

Description

Examples

References

Description

• 

Elliptic integrals are integrals of the form

abRx,yⅆx

• 

with R a rational function and y a polynomial of degree 3 or 4. This is the algebraic form of an elliptic integral. There are also trig forms (rational functions of sin and cos and a square root of a quadratic polynomial in sin and cos) and hyperbolic trig forms.

• 

Elliptic integrals are reduced to their Legendre normal form in terms of elementary functions and the Elliptic functions EllipticF, EllipticE, and EllipticPi (or their complete versions).

Examples

Elementary answer

∫0121+x41x4ⅆx

2ln17228+2ln17+2282arctan17244+π28

(1)

Symbolic parameters

assume0<k&comma;k<1

&int;0kx21x21k2x2&DifferentialD;x

EllipticFk~&comma;k~k~2EllipticEk~&comma;k~k~2

(2)

Answer as sum of roots

ans&int;0141x4&plus;245x2&plus;x4&DifferentialD;x

ans_&alpha;=RootOf_Z4+2EllipticPi14&comma;_&alpha;22&comma;1216

(3)

Can evaluate to floating point:

evalfans

0.06331207100+0.I

(4)

evalfans&comma;20

0.063312071018173992738+0.I

(5)

Trig form

&int;0Pi21&plus;2sinx&DifferentialD;x

EllipticK32+EllipticF233&comma;32+4EllipticE32EllipticPi233&comma;34&comma;32

(6)

Indefinite trig form

Itrig&int;11&plus;2cosx&DifferentialD;x

Itrig23InverseJacobiAMx2&comma;2333

(7)

Check answer:

simplifycombinexItrig11&plus;2cosx&comma;trig

0

(8)

References

  

Labahn, G., and Mutrie, M. "Reduction of Elliptic Integrals to Legendre Normal Form." University of Waterloo Tech Report 97-21, Department of Computer Science, 1997.

See Also

EllipticE

EllipticF

EllipticPi