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EllipticF

Incomplete elliptic integral of the first kind

EllipticK

Complete elliptic integral of the first kind

EllipticCK

Complementary complete elliptic integral of the first kind

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

EllipticF(z, k)

EllipticK(k)

EllipticCK(k)

Parameters

z

-

algebraic expression (the sine of the amplitude)

k

-

algebraic expression (the parameter)

Description

  

EllipticF is the Incomplete Elliptic integral of the first kind and is defined by

FunctionAdvisor(definition, EllipticF);

EllipticFz,k=∫0z1_α12+1_α12k2+1ⅆ_α1,with no restrictions on z,k

(1)
  

EllipticK and EllipticCK are respectively the Complete and the Complementary Elliptic integrals of the first kind and are defined by

FunctionAdvisor( definition, EllipticK);

EllipticKk=∫011_α12+1_α12k2+1ⅆ_α1,with no restrictions on k

(2)

FunctionAdvisor( definition, EllipticCK);

EllipticCKk=∫011_α12+11+k21_α12ⅆ_α1,with no restrictions on k

(3)
  

EllipticK, EllipticCK and EllipticF are related by

FunctionAdvisor( relate, EllipticK,EllipticF);

EllipticKk=EllipticF1,k

(4)

FunctionAdvisor( relate, EllipticK,EllipticCK);

EllipticKk=EllipticCKk2+1

(5)
  

EllipticF is also identical to the InverseJacobiSN function

FunctionAdvisor(relate, EllipticF, InverseJacobiSN);

EllipticFz,k=InverseJacobiSNz,k

(6)
  

and therefore can be used to represent all the InverseJacobiPQ functions provided some restrictions on the function parameters hold.

  

Elliptic integrals and the related functions are well described in the Table of Integrals Series and Products, Gradshteyn and Ryzhik (G&R) and in the popular Handbook of Mathematical Functions edited by Abramowitz and Stegun (A&S). In A&S, these functions are expressed in terms of a parameter m, representing the square of the modulus k entering the definition of the Elliptic, JacobiPQ and InverseJacobiPQ functions in Maple and G&R. For example, the Km function shown in A&S is numerically equal to the Maple EllipticKm command.

  

It is worth noting the difference between the Legendre normal form of the Incomplete Elliptic integral of the first kind (see A&S 17.2.7), in Maple represented by EllipticF(z,k) but for the splitting of the square root in the denominator of the integrand (see definition lines above), and the normal trigonometric form of this elliptic integral (see A&S 17.2.6), in Maple represented by the InverseJacobiAM function

InverseJacobiAM(phi,k);

InverseJacobiAMφ,k

(7)

(7) = convert((7), Int);

InverseJacobiAMφ,k=∫0φ11k2sin_θ12ⅆ_θ1

(8)
  

For instance, for -Pi/2 <= phi <= Pi/2 these two forms can be related with ease by changing variables:

EllipticF(z,k);

EllipticFz&comma;k

(9)

(9) = convert((9), Int);

EllipticFz&comma;k&equals;&int;0z1_&alpha;12&plus;1_&alpha;12k2&plus;1&DifferentialD;_&alpha;1

(10)

{z=sin(phi), _alpha1=sin(_theta1)};     #  -1 <= z <= 1

_&alpha;1&equals;sin_&theta;1&comma;z&equals;sin&phi;

(11)

PDEtools[dchange]((11), (10));

EllipticFsin&phi;&comma;k&equals;&int;0arcsinsin&phi;cos_&theta;1sin_&theta;12&plus;11k2sin_&theta;12&DifferentialD;_&theta;1

(12)

simplify((12)) assuming phi in RealRange(-Pi/2, Pi/2);

EllipticFsin&phi;&comma;k&equals;&int;0&phi;1cos_&theta;12k2k2&plus;1&DifferentialD;_&theta;1

(13)
  

where the right-hand side is actually equal to the trigonometric form InverseJacobiAMphi&comma;k. The general relationship between these two forms and the restriction on the values of the parameters such that the relation is valid are given by

FunctionAdvisor( specialize, InverseJacobiAM, EllipticF);

InverseJacobiAM&phi;&comma;k&equals;EllipticFsin&pi;floor12&real;&phi;&pi;&plus;&phi;&comma;k2floor12&real;&phi;&pi;EllipticKk&comma;with no restrictions on &phi;&comma;k&comma;InverseJacobiAM&phi;&comma;k&equals;EllipticFsin&phi;&comma;k&comma;π2<φ<π2orπ2=φAnd0φorπ2=φAndφ0

(14)

FunctionAdvisor( specialize, EllipticF, InverseJacobiAM);

EllipticFz&comma;k&equals;InverseJacobiAMarcsinz&comma;k&comma;with no restrictions on z&comma;k

(15)

Examples

Reflection symmetry and special values for EllipticK and EllipticF

FunctionAdvisorspecial_values&comma;EllipticK

EllipticKk&equals;EllipticKk&comma;EllipticK0&equals;12&pi;&comma;EllipticK&infin;&equals;0&comma;EllipticK&infin;I&equals;0

(16)

FunctionAdvisorspecial_values&comma;EllipticF

EllipticF0&comma;k&equals;0&comma;EllipticF1&comma;k&equals;EllipticKk&comma;EllipticFz&comma;0&equals;arcsinz&comma;EllipticFz&comma;1&equals;arctanhz&comma;EllipticFz&comma;&infin;&equals;0&comma;EllipticFz&comma;&infin;&equals;0

(17)

Branch points for EllipticF

FunctionAdvisorbranch_points&comma;EllipticF

EllipticFz&comma;k&comma;z&in;1&comma;1&comma;1k&comma;1k&comma;&infin;&plus;&infin;I

(18)

Branch points and the branch cut for EllipticK

FunctionAdvisorbranch_points&comma;EllipticK

EllipticKk&comma;k&in;&infin;&infin;I&comma;1&comma;1&comma;&infin;&plus;&infin;I

(19)

FunctionAdvisorbranch_cuts&comma;EllipticK

EllipticKk&comma;k<1&comma;1<k

(20)

For Rek in the cut, so for 1<=Rek<=infinity, EllipticK is continuous from below.

EllipticK21I100000

EllipticK21100000I

(21)

&equals;evalf

EllipticK21100000I&equals;0.84287832891.078252932I

(22)

EllipticK2

EllipticK2

(23)

&equals;evalf

EllipticK2&equals;0.84287517741.078257824I

(24)

EllipticK2&plus;1I100000

EllipticK2&plus;1100000I

(25)

&equals;evalf

EllipticK2&plus;1100000I&equals;0.8428783289&plus;1.078252932I

(26)

See Also

EllipticCE

EllipticCPi

EllipticE

EllipticPi

FunctionAdvisor

InverseJacobiAM

JacobiAM

RealRange

WeierstrassP

 


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