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decuslib20-02
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decus/20-0026/jelf.doc
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SUBROUTINE JELF
PURPOSE
COMPUTES THE THREE JACOBIAN ELLIPTIC FUNCTIONS SN, CN, DN.
USAGE
CALL JELF(SN,CN,DN,X,SCK)
DESCRIPTION OF PARAMETERS
SN - RESULT VALUE SN(X)
CN - RESULT VALUE CN(X)
DN - RESULT VALUE DN(X)
X - ARGUMENT OF JACOBIAN ELLIPTIC FUNCTIONS
SCK - SQUARE OF COMPLEMENTARY MODULUS
REMARKS
NONE
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
NONE
METHOD
DEFINITION
X=INTEGRAL(1/SQRT((1-T*T)*(1-(K*T)**2)), SUMMED OVER
T FROM 0 TO SN), WHERE K=SQRT(1-SCK).
SN*SN + CN*CN = 1
(K*SN)**2 + DN**2 = 1.
EVALUATION
CALCULATION IS DONE USING THE PROCESS OF THE ARITHMETIC
GEOMETRIC MEAN TOGETHER WITH GAUSS DESCENDING TRANSFORMATION
BEFORE INVERSION OF THE INTEGRAL TAKES PLACE.
REFERENCE
R. BULIRSCH, NUMERICAL CALCULATION OF ELLIPTIC INTEGRALS AND
ELLIPTIC FUNCTIOMS.
HANDBOOK SERIES OF SPECIAL FUNCTIONS
NUMERISCHE MATHEMATIK VOL. 7, 1965, PP. 78-90.