Category C. Elementary and special functions (search also class L5)

C1. Integer-valued functions C2. Powers, roots, reciprocals C3. Polynomials C4. Elementary transcendental functions C5. Exponential and logarithmic integrals C7. Gamma C8. Error functions C9. Legendre functions C10. Bessel functions C11. Confluent hypergeometric functions C14. Elliptic integrals C19. Other special functions FUNDOC-A Documentation for FNLIB, a collection of routines for evaluating elementary and special functions. C1. Integer-valued functions (e.g., floor, ceiling, factorial, binomial coefficient) BINOM-S Compute the binomial coefficients. DBINOM-D FAC-S Compute the factorial function. DFAC-D POCH-S Evaluate a generalization of Pochhammer's symbol. DPOCH-D POCH1-S Calculate a generalization of Pochhammer's symbol starting DPOCH1-D from first order. C2. Powers, roots, reciprocals CBRT-S Compute the cube root. DCBRT-D CCBRT-C C3. Polynomials C3A. Orthogonal C3A2. Chebyshev, Legendre CSEVL-S Evaluate a Chebyshev series. DCSEVL-D INITS-S Determine the number of terms needed in an orthogonal INITDS-D polynomial series so that it meets a specified accuracy. QMOMO-S This routine computes modified Chebyshev moments. The K-th DQMOMO-D modified Chebyshev moment is defined as the integral over (-1,1) of W(X)*T(K,X), where T(K,X) is the Chebyshev polynomial of degree K. XLEGF-S Compute normalized Legendre polynomials and associated DXLEGF-D Legendre functions. XNRMP-S Compute normalized Legendre polynomials. DXNRMP-D C4. Elementary transcendental functions C4A. Trigonometric, inverse trigonometric CACOS-C Compute the complex arc cosine. CASIN-C Compute the complex arc sine. CATAN-C Compute the complex arc tangent. CATAN2-C Compute the complex arc tangent in the proper quadrant. COSDG-S Compute the cosine of an argument in degrees. DCOSDG-D COT-S Compute the cotangent. DCOT-D CCOT-C CTAN-C Compute the complex tangent. SINDG-S Compute the sine of an argument in degrees. DSINDG-D C4B. Exponential, logarithmic ALNREL-S Evaluate ln(1+X) accurate in the sense of relative error. DLNREL-D CLNREL-C CLOG10-C Compute the principal value of the complex base 10 logarithm. EXPREL-S Calculate the relative error exponential (EXP(X)-1)/X. DEXPRL-D CEXPRL-C C4C. Hyperbolic, inverse hyperbolic ACOSH-S Compute the arc hyperbolic cosine. DACOSH-D CACOSH-C ASINH-S Compute the arc hyperbolic sine. DASINH-D CASINH-C ATANH-S Compute the arc hyperbolic tangent. DATANH-D CATANH-C CCOSH-C Compute the complex hyperbolic cosine. CSINH-C Compute the complex hyperbolic sine. CTANH-C Compute the complex hyperbolic tangent. C5. Exponential and logarithmic integrals ALI-S Compute the logarithmic integral. DLI-D E1-S Compute the exponential integral E1(X). DE1-D EI-S Compute the exponential integral Ei(X). DEI-D EXINT-S Compute an M member sequence of exponential integrals DEXINT-D E(N+K,X), K=0,1,...,M-1 for N .GE. 1 and X .GE. 0. SPENC-S Compute a form of Spence's integral due to K. Mitchell. DSPENC-D C7. Gamma C7A. Gamma, log gamma, reciprocal gamma ALGAMS-S Compute the logarithm of the absolute value of the Gamma DLGAMS-D function. ALNGAM-S Compute the logarithm of the absolute value of the Gamma DLNGAM-D function. CLNGAM-C C0LGMC-C Evaluate (Z+0.5)*LOG((Z+1.)/Z) - 1.0 with relative accuracy. GAMLIM-S Compute the minimum and maximum bounds for the argument in DGAMLM-D the Gamma function. GAMMA-S Compute the complete Gamma function. DGAMMA-D CGAMMA-C GAMR-S Compute the reciprocal of the Gamma function. DGAMR-D CGAMR-C POCH-S Evaluate a generalization of Pochhammer's symbol. DPOCH-D POCH1-S Calculate a generalization of Pochhammer's symbol starting DPOCH1-D from first order. C7B. Beta, log beta ALBETA-S Compute the natural logarithm of the complete Beta DLBETA-D function. CLBETA-C BETA-S Compute the complete Beta function. DBETA-D CBETA-C C7C. Psi function PSI-S Compute the Psi (or Digamma) function. DPSI-D CPSI-C PSIFN-S Compute derivatives of the Psi function. DPSIFN-D C7E. Incomplete gamma GAMI-S Evaluate the incomplete Gamma function. DGAMI-D GAMIC-S Calculate the complementary incomplete Gamma function. DGAMIC-D GAMIT-S Calculate Tricomi's form of the incomplete Gamma function. DGAMIT-D C7F. Incomplete beta BETAI-S Calculate the incomplete Beta function. DBETAI-D C8. Error functions C8A. Error functions, their inverses, integrals, including the normal distribution function ERF-S Compute the error function. DERF-D ERFC-S Compute the complementary error function. DERFC-D C8C. Dawson's integral DAWS-S Compute Dawson's function. DDAWS-D C9. Legendre functions XLEGF-S Compute normalized Legendre polynomials and associated DXLEGF-D Legendre functions. XNRMP-S Compute normalized Legendre polynomials. DXNRMP-D C10. Bessel functions C10A. J, Y, H-(1), H-(2) C10A1. Real argument, integer order BESJ0-S Compute the Bessel function of the first kind of order DBESJ0-D zero. BESJ1-S Compute the Bessel function of the first kind of order one. DBESJ1-D BESY0-S Compute the Bessel function of the second kind of order DBESY0-D zero. BESY1-S Compute the Bessel function of the second kind of order DBESY1-D one. C10A3. Real argument, real order BESJ-S Compute an N member sequence of J Bessel functions DBESJ-D J/SUB(ALPHA+K-1)/(X), K=1,...,N for non-negative ALPHA and X. BESY-S Implement forward recursion on the three term recursion DBESY-D relation for a sequence of non-negative order Bessel functions Y/SUB(FNU+I-1)/(X), I=1,...,N for real, positive X and non-negative orders FNU. C10A4. Complex argument, real order CBESH-C Compute a sequence of the Hankel functions H(m,a,z) ZBESH-C for superscript m=1 or 2, real nonnegative orders a=b, b+1,... where b>0, and nonzero complex argument z. A scaling option is available to help avoid overflow. CBESJ-C Compute a sequence of the Bessel functions J(a,z) for ZBESJ-C complex argument z and real nonnegative orders a=b,b+1, b+2,... where b>0. A scaling option is available to help avoid overflow. CBESY-C Compute a sequence of the Bessel functions Y(a,z) for ZBESY-C complex argument z and real nonnegative orders a=b,b+1, b+2,... where b>0. A scaling option is available to help avoid overflow. C10B. I, K C10B1. Real argument, integer order BESI0-S Compute the hyperbolic Bessel function of the first kind DBESI0-D of order zero. BESI0E-S Compute the exponentially scaled modified (hyperbolic) DBSI0E-D Bessel function of the first kind of order zero. BESI1-S Compute the modified (hyperbolic) Bessel function of the DBESI1-D first kind of order one. BESI1E-S Compute the exponentially scaled modified (hyperbolic) DBSI1E-D Bessel function of the first kind of order one. BESK0-S Compute the modified (hyperbolic) Bessel function of the DBESK0-D third kind of order zero. BESK0E-S Compute the exponentially scaled modified (hyperbolic) DBSK0E-D Bessel function of the third kind of order zero. BESK1-S Compute the modified (hyperbolic) Bessel function of the DBESK1-D third kind of order one. BESK1E-S Compute the exponentially scaled modified (hyperbolic) DBSK1E-D Bessel function of the third kind of order one. C10B3. Real argument, real order BESI-S Compute an N member sequence of I Bessel functions DBESI-D I/SUB(ALPHA+K-1)/(X), K=1,...,N or scaled Bessel functions EXP(-X)*I/SUB(ALPHA+K-1)/(X), K=1,...,N for non-negative ALPHA and X. BESK-S Implement forward recursion on the three term recursion DBESK-D relation for a sequence of non-negative order Bessel functions K/SUB(FNU+I-1)/(X), or scaled Bessel functions EXP(X)*K/SUB(FNU+I-1)/(X), I=1,...,N for real, positive X and non-negative orders FNU. BESKES-S Compute a sequence of exponentially scaled modified Bessel DBSKES-D functions of the third kind of fractional order. BESKS-S Compute a sequence of modified Bessel functions of the DBESKS-D third kind of fractional order. C10B4. Complex argument, real order CBESI-C Compute a sequence of the Bessel functions I(a,z) for ZBESI-C complex argument z and real nonnegative orders a=b,b+1, b+2,... where b>0. A scaling option is available to help avoid overflow. CBESK-C Compute a sequence of the Bessel functions K(a,z) for ZBESK-C complex argument z and real nonnegative orders a=b,b+1, b+2,... where b>0. A scaling option is available to help avoid overflow. C10D. Airy and Scorer functions AI-S Evaluate the Airy function. DAI-D AIE-S Calculate the Airy function for a negative argument and an DAIE-D exponentially scaled Airy function for a non-negative argument. BI-S Evaluate the Bairy function (the Airy function of the DBI-D second kind). BIE-S Calculate the Bairy function for a negative argument and an DBIE-D exponentially scaled Bairy function for a non-negative argument. CAIRY-C Compute the Airy function Ai(z) or its derivative dAi/dz ZAIRY-C for complex argument z. A scaling option is available to help avoid underflow and overflow. CBIRY-C Compute the Airy function Bi(z) or its derivative dBi/dz ZBIRY-C for complex argument z. A scaling option is available to help avoid overflow. C10F. Integrals of Bessel functions BSKIN-S Compute repeated integrals of the K-zero Bessel function. DBSKIN-D C11. Confluent hypergeometric functions CHU-S Compute the logarithmic confluent hypergeometric function. DCHU-D C14. Elliptic integrals RC-S Calculate an approximation to DRC-D RC(X,Y) = Integral from zero to infinity of -1/2 -1 (1/2)(t+X) (t+Y) dt, where X is nonnegative and Y is positive. RD-S Compute the incomplete or complete elliptic integral of the DRD-D 2nd kind. For X and Y nonnegative, X+Y and Z positive, RD(X,Y,Z) = Integral from zero to infinity of -1/2 -1/2 -3/2 (3/2)(t+X) (t+Y) (t+Z) dt. If X or Y is zero, the integral is complete. RF-S Compute the incomplete or complete elliptic integral of the DRF-D 1st kind. For X, Y, and Z non-negative and at most one of them zero, RF(X,Y,Z) = Integral from zero to infinity of -1/2 -1/2 -1/2 (1/2)(t+X) (t+Y) (t+Z) dt. If X, Y or Z is zero, the integral is complete. RJ-S Compute the incomplete or complete (X or Y or Z is zero) DRJ-D elliptic integral of the 3rd kind. For X, Y, and Z non- negative, at most one of them zero, and P positive, RJ(X,Y,Z,P) = Integral from zero to infinity of -1/2 -1/2 -1/2 -1 (3/2)(t+X) (t+Y) (t+Z) (t+P) dt. C19. Other special functions RC3JJ-S Evaluate the 3j symbol f(L1) = ( L1 L2 L3) DRC3JJ-D (-M2-M3 M2 M3) for all allowed values of L1, the other parameters being held fixed. RC3JM-S Evaluate the 3j symbol g(M2) = (L1 L2 L3 ) DRC3JM-D (M1 M2 -M1-M2) for all allowed values of M2, the other parameters being held fixed. RC6J-S Evaluate the 6j symbol h(L1) = {L1 L2 L3} DRC6J-D {L4 L5 L6} for all allowed values of L1, the other parameters being held fixed.