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decus/20-0026/eli1.doc
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SUBROUTINE ELI1
PURPOSE
COMPUTES THE ELLIPTIC INTEGRAL OF FIRST KIND
USAGE
CALL ELI1(RES,X,CK)
DESCRIPTION OF PARAMETERS
RES - RESULT VALUE
X - UPPER INTEGRATION BOUND (ARGUMENT OF ELLIPTIC
INTEGRAL OF FIRST KIND)
CK - COMPLEMENTARY MODULUS
REMARKS
MODULUS K = SQRT(1.-CK*CK).
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
NONE
METHOD
DEFINITION
RES=INTEGRAL(1/SQRT((1+T*T)*(1+(CK*T)**2)), SUMMED
OVER T FROM 0 TO X).
EQUIVALENT ARE THE DEFINITIONS
RES=INTEGRAL(1/(COS(T)*SQRT(1+(CK*TAN(T))**2)), SUMMED
OVER T FROM 0 TO ATAN(X)),
RES=INTEGRAL(1/SQRT(1-(K*SIN(T))**2), SUMMED OVER
T FROM 0 TO ATAN(X)).
EVALUATION
LANDENS TRANSFORMATION IS USED FOR CALCULATION.
REFERENCE
R. BULIRSCH, NUMERICAL CALCULATION OF ELLIPTIC INTEGRALS AND
ELLIPTIC FUNCTIONS.
HANDBOOK SERIES OF SPECIAL FUNCTIONS
NUMERISCHE MATHEMATIK VOL. 7, 1965, PP. 78-90.