*DECK FUNDOC SUBROUTINE FUNDOC C***BEGIN PROLOGUE FUNDOC C***PURPOSE Documentation for FNLIB, a collection of routines for C evaluating elementary and special functions. C***LIBRARY SLATEC C***CATEGORY C, Z C***TYPE ALL (FUNDOC-A) C***KEYWORDS DOCUMENTATION, ELEMENTARY FUNCTIONS, SPECIAL FUNCTIONS C***AUTHOR Kahaner, D. K., (NBS) C***DESCRIPTION C C The SLATEC Library -- Elementary And Special Functions C C This describes the elementary and special function routines available C in the SLATEC library. Most of the these routines were written by C Wayne Fullerton while at LANL. Some were written by Don Amos of SNLA. C There are approximately 63 single precision, 63 double precision and C 25 complex user callable elementary and special function routines. C C The table below gives a breakdown of routines according to their C function. Unless otherwise indicated all routines are function C subprograms. C Sngl. Dble. C Description Notation Prec. Prec. Complex C C ***Intrinsic Functions and Fundamental Functions*** C Unpack floating point Call R9UPAK(X,Y,N) D9UPAK -- C number C Pack floating point R9PAK(Y,N) D9PAK -- C number C Initialize orthogonal INITS(OS,NOS,ETA) INITDS -- C polynomial series C Evaluate Chebyshev summation for CSEVL(X,CS,N) DCSEVL -- C series i = 1 to n of C cs(i)*(2*x)**(i-1) C C ***Elementary Functions*** C Argument = theta in z = \ z \ * -- -- CARG(Z) C radians e**(i * theta) C Cube root CBRT(X) DCBRT CCBRT C Relative error exponen- ((e**x) -1) / x EXPREL(X) DEXPRL CEXPRL C tial from first order C Common logarithm log to the base 10 -- -- CLOG10(Z) C of z C Relative error logarithm ln(1 + x) ALNREL(X) DLNREL CLNREL C Relative error logarithm (ln(1 + x) - x R9LN2R(X) D9LN2R C9LN2R C from second order + x**2/2) / x**3 C ***Trigonometric and Hyperbolic Functions*** C Tangent tan z -- -- CTAN(Z) C Cotangent cot x COT(X) DCOT CCOT C Sine x in degrees sin((2*pi*x)/360) SINDG(X) DSINDG -- C Cosine x in degrees cos((2*pi*x)/360) COSDG(X) DCOSDG -- C Arc sine arcsin (z) -- -- CASIN(Z) C Arc cosine arccos (z) -- -- CACOS(Z) C Arc tangent arctan (z) -- -- CATAN(Z) C Quadrant correct arctan (z1/z2) -- -- CATAN2(Z1, C arc tangent Z2) C Hyperbolic sine sinh z -- -- CSINH(Z) C Hyperbolic cosine cosh z -- -- CCOSH(Z) C Hyperbolic tangent tanh z -- -- CTANH(Z) C Arc hyperbolic sine arcsinh (x) ASINH(X) DASINH CASINH C Arc hyperbolic cosine arccosh (x) ACOSH(X) DACOSH CACOSH C Arc hyperbolic tangent arctanh (x) ATANH(X) DATANH CATANH C Relative error arc (arctan (x) - x) R9ATN1(X) D9ATN1 -- C tangent from first order / x**3 C ***Exponential Integrals and Related Functions*** C Exponential integral Ei(x) = (minus) EI(X) DEI -- C the integral from C -x to infinity of C (e**-t / t)dt C Exponential integral E sub 1 (x) = E1(X) DE1 -- C the integral from x C to infinity of C (e**-t / t) dt C Logarithmic integral li(x) = the ALI(X) DLI -- C integral from 0 to C x of (1 / ln t) dt C Sequences of exponential integrals. C M values are computed where C k=0,1,...M-1 and n>=1 C Exponential integral E sub n+k (x) Call EXINT(X, DEXINT -- C =the integral from N,KODE,M,TOL, C 1 to infinity of EN,IERR) C (e**(-x*t)/t**(n+k))dt C ***Gamma Functions and Related Functions*** C Factorial n! FAC(N) DFAC -- C Binomial n!/(m!*(n-m)!) BINOM(N,M) DBINOM -- C Gamma gamma(x) GAMMA(X) DGAMMA CGAMMA C Gamma(x) under and Call GAMLIM( DGAMLM -- C overflow limits XMIN,XMAX) C Reciprocal gamma 1 / gamma(x) GAMR(X) DGAMR CGAMR C Log abs gamma ln \gamma(x)\ ALNGAM(X) DLNGAM -- C Log gamma ln gamma(z) -- -- CLNGAM C Log abs gamma g = ln \gamma(x)\ Call ALGAMS(X, DLGAMS -- C with sign s = sign gamma(x) G,S) C Incomplete gamma gamma(a,x) = GAMI(A,X) DGAMI -- C the integral from C 0 to x of C (t**(a-1) * e**-t)dt C Complementary gamma(a,x) = GAMIC(A,X) DGAMIC -- C incomplete gamma the integral from C x to infinity of C (t**(a-1) * e**-t)dt C Tricomi's gamma super star(a,x) GAMIT(A,X) DGAMIT -- C incomplete gamma = x**-a * C incomplete gamma(a,x) C / gamma(a) C Psi (Digamma) psi(x) = gamma'(x) PSI(X) DPSI CPSI C / gamma(x) C Pochhammer's (a) sub x = gamma(a+x) POCH(A,X) DPOCH -- C generalized symbol / gamma(a) C Pochhammer's symbol ((a) sub x -1) / x POCH1(A,X) DPOCH1 -- C from first order C Beta b(a,b) = (gamma(a) BETA(A,B) DBETA CBETA C * gamma(b)) C / gamma(a+b) C = the integral C from 0 to 1 of C (t**(a-1) * C (1-t)**(b-1))dt C Log beta ln b(a,b) ALBETA(A,B) DLBETA CLBETA C Incomplete beta i sub x (a,b) = BETAI(X,A,B) DBETAI __ C b sub x (a,b) / b(a,b) C = 1 / b(a,b) * C the integral C from 0 to x of C (t**(a-1) * C (1-t)**(b-1))dt C Log gamma correction ln gamma(x) - R9LGMC(X) D9LGMC C9LGMC C term when Stirling's (ln(2 * pi))/2 - C approximation is valid (x - 1/2) * ln(x) + x C ***Error Functions and Fresnel Integrals*** C Error function erf x = (2 / ERF(X) DERF -- C square root of pi) * C the integral from C 0 to x of C e**(-t**2)dt C Complementary erfc x = (2 / ERFC(X) DERFC -- C error function square root of pi) * C the integral from C x to infinity of C e**(-t**2)dt C Dawson's function F(x) = e**(-x**2) DAWS(X) DDAWS -- C * the integral from C from 0 to x of C e**(t**2)dt C ***Bessel Functions*** C Bessel functions of special integer order C First kind, order zero J sub 0 (x) BESJ0(X) DBESJ0 -- C First kind, order one J sub 1 (x) BESJ1(X) DBESJ1 -- C Second kind, order zero Y sub 0 (x) BESY0(X) DBESY0 -- C Second kind, order one Y sub 1 (x) BESY1(X) DBESY1 -- C Modified (hyperbolic) Bessel functions of special integer order C First kind, order zero I sub 0 (x) BESI0(X) DBESI0 -- C First kind, order one I sub 1 (x) BESI1(X) DBESI1 -- C Third kind, order zero K sub 0 (x) BESK0(X) DBESK0 -- C Third kind, order one K sub 1 (x) BESK1(X) DBESK1 -- C Modified (hyperbolic) Bessel functions of special integer order C scaled by an exponential C First kind, order zero e**-\x\ * I sub 0(x) BESI0E(X) DBSI0E -- C First kind, order one e**-\x\ * I sub 1(x) BESI1E(X) DBSI1E -- C Third kind, order zero e**x * K sub 0 (x) BESK0E(X) DBSK0E -- C Third kind, order one e**x * K sub 1 (x) BESK1E(X) DBSK1E -- C Sequences of Bessel functions of general order. C N values are computed where k = 1,2,...N and v .ge. 0. C Modified first kind I sub v+k-1 (x) Call BESI(X, DBESI -- C optional scaling ALPHA,KODE,N, C by e**(-x) Y,NZ) C First kind J sub v+k-1 (x) Call BESJ(X, DBESJ -- C ALPHA,N,Y,NZ) C Second kind Y sub v+k-1 (x) Call BESY(X, DBESY -- C FNU,N,Y) C Modified third kind K sub v+k-1 (x) Call BESK(X, DBESK -- C optional scaling FNU,KODE,N,Y, C by e**(x) NZ) C Sequences of Bessel functions. \N\ values are computed where C I = 0, 1, 2, ..., N-1 for N > 0 or I = 0, -1, -2, ..., N+1 C for N < 0. C Modified third kind K sub v+i (x) Call BESKS( DBESKS -- C XNU,X,N,BK) C Sequences of Bessel functions scaled by an exponential. C \N\ values are computed where I = 0, 1, 2, ..., N-1 C for N > 0 or I = 0, -1, -2, ..., N+1 for N < 0. C Modified third kind e**x * Call BESKES( DBSKES -- C K sub v+i (x) XNU,X,N,BK) C ***Bessel Functions of Fractional Order*** C Airy functions C Airy Ai(x) AI(X) DAI -- C Bairy Bi(x) BI(X) DBI -- C Exponentially scaled Airy functions C Airy Ai(x), x <= 0 AIE(X) DAIE -- C exp(2/3 * x**(3/2)) C * Ai(x), x >= 0 C Bairy Bi(x), x <= 0 BIE(X) DBIE -- C exp(-2/3 * x**(3/2)) C * Bi(x), x >= 0 C ***Confluent Hypergeometric Functions*** C Confluent U(a,b,x) CHU(A,B,X) DCHU -- C hypergeometric C ***Miscellaneous Functions*** C Spence s(x) = - the SPENC(X) DSPENC -- C dilogarithm integral from C 0 to x of C ((ln \1-y\) / y)dy C C***REFERENCES (NONE) C***ROUTINES CALLED (NONE) C***REVISION HISTORY (YYMMDD) C 801015 DATE WRITTEN C 861211 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900326 Routine name changed from FNLIBD to FUNDOC. (WRB) C 900723 PURPOSE section revised. (WRB) C***END PROLOGUE FUNDOC C***FIRST EXECUTABLE STATEMENT FUNDOC RETURN END