*DECK C9LN2R COMPLEX FUNCTION C9LN2R (Z) C***BEGIN PROLOGUE C9LN2R C***SUBSIDIARY C***PURPOSE Evaluate LOG(1+Z) from second order relative accuracy so C that LOG(1+Z) = Z - Z**2/2 + Z**3*C9LN2R(Z). C***LIBRARY SLATEC (FNLIB) C***CATEGORY C4B C***TYPE COMPLEX (R9LN2R-S, D9LN2R-D, C9LN2R-C) C***KEYWORDS ELEMENTARY FUNCTIONS, FNLIB, LOGARITHM, SECOND ORDER C***AUTHOR Fullerton, W., (LANL) C***DESCRIPTION C C Evaluate LOG(1+Z) from 2-nd order with relative error accuracy so C that LOG(1+Z) = Z - Z**2/2 + Z**3*C9LN2R(Z). C C Now LOG(1+Z) = 0.5*LOG(1+2*X+ABS(Z)**2) + I*CARG(1+Z), C where X = REAL(Z) and Y = AIMAG(Z). C We find C Z**3 * C9LN2R(Z) = -X*ABS(Z)**2 - 0.25*ABS(Z)**4 C + (2*X+ABS(Z)**2)**3 * R9LN2R(2*X+ABS(Z)**2) C + I * (CARG(1+Z) + (X-1)*Y) C The imaginary part must be evaluated carefully as C (ATAN(Y/(1+X)) - Y/(1+X)) + Y/(1+X) - (1-X)*Y C = (Y/(1+X))**3 * R9ATN1(Y/(1+X)) + X**2*Y/(1+X) C C Now we divide through by Z**3 carefully. Write C 1/Z**3 = (X-I*Y)/ABS(Z)**3 * (1/ABS(Z)**3) C then C9LN2R(Z) = ((X-I*Y)/ABS(Z))**3 * (-X/ABS(Z) - ABS(Z)/4 C + 0.5*((2*X+ABS(Z)**2)/ABS(Z))**3 * R9LN2R(2*X+ABS(Z)**2) C + I*Y/(ABS(Z)*(1+X)) * ((X/ABS(Z))**2 + C + (Y/(ABS(Z)*(1+X)))**2 * R9ATN1(Y/(1+X)) ) ) C C If we let XZ = X/ABS(Z) and YZ = Y/ABS(Z) we may write C C9LN2R(Z) = (XZ-I*YZ)**3 * (-XZ - ABS(Z)/4 C + 0.5*(2*XZ+ABS(Z))**3 * R9LN2R(2*X+ABS(Z)**2) C + I*YZ/(1+X) * (XZ**2 + (YZ/(1+X))**2*R9ATN1(Y/(1+X)) )) C C***REFERENCES (NONE) C***ROUTINES CALLED R9ATN1, R9LN2R C***REVISION HISTORY (YYMMDD) C 780401 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900720 Routine changed from user-callable to subsidiary. (WRB) C***END PROLOGUE C9LN2R COMPLEX Z C***FIRST EXECUTABLE STATEMENT C9LN2R X = REAL (Z) Y = AIMAG (Z) C CABSZ = ABS(Z) IF (CABSZ.GT.0.8125) GO TO 20 C C9LN2R = CMPLX (1.0/3.0, 0.0) IF (CABSZ.EQ.0.0) RETURN C XZ = X/CABSZ YZ = Y/CABSZ C ARG = 2.0*XZ + CABSZ RPART = 0.5*ARG**3*R9LN2R(CABSZ*ARG) - XZ - 0.25*CABSZ Y1X = YZ/(1.0+X) AIPART = Y1X * (XZ**2 + Y1X**2*R9ATN1(CABSZ*Y1X) ) C C9LN2R = CMPLX(XZ,-YZ)**3 * CMPLX(RPART,AIPART) RETURN C 20 C9LN2R = (LOG(1.0+Z) - Z*(1.0-0.5*Z)) / Z**3 RETURN C END