IMSL MATH/LIBRARY Special Functions

C....... ELEMENTARY AND SPECIAL FUNCTIONS (search also class L5)

C1..... Integer-valued functions (e.g., floor, ceiling, factorial, binomial coefficient)

BINOM   Evaluate the binomial coefficient.

FAC     Evaluate the factorial of the argument.

C2..... Powers, roots, reciprocals

CBRT   Evaluate the cube root.

CCBRT  Evaluate the complex cube root.

C3..... Polynomials

C3a.... Orthogonal

INITS   Initialize the orthogonal series so the function value is the number of terms needed to insure the error is no larger than the requested accuracy.

C3a2.. Chebyshev, Legendre

CSEVL   Evaluate the N-term Chebyshev series.

C4..... Elementary transcendental functions

C4a     Trigonometric, inverse trigonometric

CACOS   Evaluate the complex arc cosine.

CARG    Evaluate the argument of a complex number.

CASIN   Evaluate the complex arc sine.

CATAN   Evaluate the complex arc tangent.

CATAN2 Evaluate the complex arc tangent of a ratio.

CCOT    Evaluate the complex cotangent.

COSDG   Evaluate the cosine for the argument in degrees.

COT    Evaluate the cotangent.

SINDG  Evaluate the sine for the argument in degrees.

C4b.... Exponential, logarithmic

ALNREL Evaluate the natural logarithm of one plus the argument.

CEXPRL Evaluate the complex exponential function factored from first order.

CLNREL Evaluate the principal value of the complex natural logarithm of one plus the argument.

CLOG10 Evaluate the principal value of the complex common logarithm.

EXPRL   Evaluate the exponential function factored from first order, (EXP(X) - 1.0)/X.

C4c.... Hyperbolic, inverse hyperbolic

ACOSH   Evaluate the arc hyperbolic cosine.

ASINH   Evaluate the arc hyperbolic sine.

ATANH   Evaluate the arc hyperbolic tangent.

CACOSH  Evaluate the complex arc hyperbolic cosine.

CASINH  Evaluate the complex arc hyperbolic sine.

CATANH  Evaluate the complex arc hyperbolic tangent.

CCOSH   Evaluate the complex hyperbolic cosine.

CSINH   Evaluate the complex hyperbolic sine.

CTAN     Evaluate the complex tangent.

CTANH   Evaluate the complex hyperbolic tangent.

C5..... Exponential and logarithmic integrals

ALI     Evaluate the logarithmic integral.

CHI       Evaluate the hyperbolic cosine integral.

CI         Evaluate the cosine integral.

CIN       Evaluate a function closely related to the cosine integral.

CINH     Evaluate a function closely related to the hyperbolic cosine integral.

E1         Evaluate the exponential integral for arguments greater than zero and the Cauchy principal value of the integral for arguments less than zero.

EI         Evaluate the exponential integral for arguments greater than zero and the Cauchy principal value for arguments less than zero.

ENE       Evaluate the exponential integral of integer order for arguments greater than zero scaled by EXP(X).

SHI       Evaluate the hyperbolic sine integral.

SI         Evaluate the sine integral.

C7..... Gamma

C7a.... Gamma, log gamma, reciprocal gamma

ALGAMS Return the logarithm of the absolute value of the gamma function and the sign of gamma.

ALNGAM Evaluate the logarithm of the absolute value of the gamma function.

CGAMMA Evaluate the complex gamma function.

CGAMR   Evaluate the reciprocal complex gamma function.

CLNGAM Evaluate the complex natural logarithm of the gamma function.

GAMMA   Evaluate the complete gamma function.

GAMR    Evaluate the reciprocal gamma function.

POCH    Evaluate a generalization of Pochhammer's symbol.

POCH1   Evaluate a generalization of Pochhammer's symbol starting from the first order.

C7b.... Beta, log beta

ALBETA Evaluate the natural logarithm of the complete beta function for positive arguments.

BETA    Evaluate the complete beta function.

CBETA   Evaluate the complex complete beta function.

CLBETA Evaluate the complex logarithm of the complete beta function.

C7c.... Psi function

CPSI    Evaluate the logarithmic derivative of the gamma function for a complex argument.

PSI     Evaluate the logarithmic derivative of the gamma function.

C7e.... Incomplete gamma

CHIDF   Evaluate the chi-squared distribution function.

CHIIN   Evaluate the inverse of the chi-squared distribution function.

GAMDF   Evaluate the gamma distribution function.

GAMI    Evaluate the incomplete gamma function.

GAMIC   Evaluate the complementary incomplete gamma function.

GAMIT   Evaluate the Tricomi form of the incomplete gamma function.

C7f.... Incomplete beta

BETAI   Evaluate the incomplete beta function ratio.

BETDF   Evaluate the beta probability distribution function.

BETIN   Evaluate the inverse of the beta distribution function.

C8..... Error functions

C8a.... Error functions, their inverses, integrals, including the normal distribution function

ANORDF Evaluate the standard normal (Gaussian) distribution function.

ANORIN Evaluate the inverse of the standard normal (Gaussian) distribution function.

CERFE   Evaluate the complex scaled complemented error function.

ERF     Evaluate the error function.

ERFC    Evaluate the complementary error function.

ERFCE   Evaluate the exponentially scaled complementary error function.

ERFCI   Evaluate the inverse complementary error function.

ERFI    Evaluate the inverse error function.

C8b.... Fresnel integrals

FRESC   Evaluate the cosine Fresnel integral.

FRESS   Evaluate the sine Fresnel integral.

C8c.... Dawson's integral

DAWS  Evaluate Dawson function.

C10.... Bessel functions

C10a.. J, Y, H(1); H(2)

C10a1 Real argument, integer order

BSJ0    Evaluate the Bessel function of the first kind of order zero.

BSJ1    Evaluate the Bessel function of the first kind of order one.

BSJNS   Evaluate a sequence of Bessel functions of the first kind with integer order and real arguments.

BSY0    Evaluate the Bessel function of the second kind of order zero.

BSY1    Evaluate the Bessel function of the second kind of order one.

C10a2 Complex argument, integer order.

CBJNS   Evaluate a sequence of Bessel functions of the first kind with integer order and complex arguments.

C10a3 Real argument, real order

BSJS    Evaluate a sequence of Bessel functions of the first kind with real order and real positive arguments.

BSYS    Evaluate a sequence of Bessel functions of the second kind with real nonnegative order and real positive arguments.

C10a4 Complex argument, real order

CBJS    Evaluate a sequence of Bessel functions of the first kind with real order and complex arguments.

CBYS    Evaluate a sequence of Bessel functions of the second kind with real order and complex arguments.

C10b.. I, K

C10b1 Real argument, integer order

BSI0    Evaluate the modified Bessel function of the first kind of order zero.

BSI0E   Evaluate the exponentially scaled modified Bessel function of the first kind of order zero.

BSI1    Evaluate the modified Bessel function of the first kind of order one.

BSI1E   Evaluate the exponentially scaled modified Bessel function of the first kind of order one.

BSINS   Evaluate a sequence of Modified Bessel functions of the first kind with integer order and real arguments.

BSK0    Evaluate the modified Bessel function of the third kind of order zero.

BSK0E   Evaluate the exponentially scaled modified Bessel function of the third kind of order zero.

BSK1    Evaluate the modified Bessel function of the third kind of order one.

BSK1E   Evaluate the exponentially scaled modified Bessel function of the third kind of order one.

C10b2 Complex argument, integer order

CBINS   Evaluate a sequence of Modified Bessel functions of the first kind with integer order and complex arguments.

C10b3 Real argument, real order

BSIES   Evaluate a sequence of exponentially scaled Modified Bessel functions of the first kind with nonnegative real order and real positive arguments.

BSIS    Evaluate a sequence of Modified Bessel functions of the first kind with real order and real positive arguments.

BSKES   Evaluate a sequence of exponentially scaled modified Bessel functions of the third kind of fractional order.

BSKS    Evaluate a sequence of modified Bessel functions of the third kind of fractional order.

C10b4 Complex argument, real order

CBIS    Evaluate a sequence of Modified Bessel functions of the first kind with real order and complex arguments.

CBKS    Evaluate a sequence of Modified Bessel functions of the second kind with real order and complex arguments.

C10c.. Kelvin functions

AKEI0   Evaluate the Kelvin function of the second kind, kei, of order zero.

AKEI1   Evaluate the Kelvin function of the second kind, kei, of order one.

AKEIP0 Evaluates the derivative of the Kelvin function of the second kind, kei, of order zero.

AKER0   Evaluate the Kelvin function of the second kind, ker, of order zero.

AKER1   Evaluate the Kelvin function of the second kind, ker, of order one.

AKERP0 Evaluate the derivative of the Kelvin function of the second kind, ker, of order zero.

BEI0    Evaluate the Kelvin function of the first kind, bei, of order zero.

BEI1    Evaluate the Kelvin function of the first kind, bei, of order one.

BEIP0   Evaluate the derivative of the Kelvin function of the first kind, bei, of order zero.

BER0    Evaluate the Kelvin function of the first kind, ber, of order zero.

BER1    Evaluate the Kelvin function of the first kind, ber, of order one.

BERP0   Evaluate the derivative of the Kelvin function of the first kind, ber, of order zero.

C10d.. Airy and Scorer functions

AI       Evaluate the Airy function.

AID     Evaluate the derivative of the Airy function.

AIDE    Evaluate the exponentially scaled derivative of the Airy

function.

AIE     Evaluate the exponentially scaled Airy function.

BI       Evaluate the Airy function of the second kind.

BID     Evaluate the derivative of the Airy function of the second kind.

BIDE    Evaluate the exponentially scaled derivative of the Airy function of the second kind.

BIE     Evaluate the exponentially scaled Airy function of the second kind.

C14.... Elliptic integrals

CEJCN   Evaluate the complex Jacobi elliptic integral cn(z, m).

CEJDN   Evaluate the complex Jacobi elliptic integral dn(z, m).

CEJSN   Evaluate the complex Jacobi elliptic function sn(z, m).

EJCN    Evaluate the Jacobi elliptic function cn(x, m).

EJDN    Evaluate the Jacobi elliptic function dn(x, m).

EJSN    Evaluate the Jacobi elliptic function sn(x, m).

ELE     Evaluate the complete elliptic integral of the second kind E(x).

ELK     Evaluate the complete elliptic integral of the kind K(x).

ELRC    Evaluate an elementary integral from which inverse circular functions, logarithms and inverse hyperbolic functions can be computed.

ELRD    Evaluate Carlson's incomplete elliptic integral of the second kind RD(X, Y, Z).

ELRF    Evaluate Carlson's incomplete elliptic integral of the first kind RF(X, Y, Z).

ELRJ    Evaluate Carlson's incomplete elliptic integral of the third kind RJ(X, Y, Z, RHO).

C15.... Weierstrass elliptic functions

CWPL    Evaluate the Weierstrass P-function in the lemniscat case for complex argument with unit period parallelogram.

CWPLD   Evaluate the first derivative of the Weierstrass P-function in the lemniscatic case for complex argum with unit period parallelogram.

CWPQ    Evaluate the Weierstrass P-function in the equianharmonic case for complex argument with unit period parallelogram.

CWPQD   Evaluate the first derivative of the Weierstrass P-function in the equianharmonic case for complex argument with unit period parallelogram.

C17.... Mathieu functions

MATCE   Evaluate a sequence of even, periodic, integer order, real Mathieu functions.

MATEE   Evaluate the eigenvalues for the periodic Mathieu functions.

MATSE   Evaluate a sequence of odd, periodic, integer order, real Mathieu functions.

C19.... Other special functions

SPENC   Evaluate a form of Spence's integral.

L....... STATISTICS, PROBABILITY

L5...... Function evaluation (search also class C)

L5a.... Univariate

L5a1.. Cumulative distribution functions, probability density functions

GCDF    Evaluate a general continuous cumulative distribution function given ordinates of the density.

L5a1b. Beta, binomial

BETDF   Evaluate the beta probability distribution function.

BINDF   Evaluate the binomial distribution function.

BINPR   Evaluate the binomial probability function.

L5a1c. Cauchy, chi-squared

CHIDF   Evaluate the chi-squared distribution function.

CSNDF   Evaluate the noncentral chi-squared distribution function.

L5a1f. F distribution

FDF     Evaluate the F distribution function.

L5a1g. Gamma, general, geometric

GAMDF   Evaluate the gamma distribution function.

L5a1h. Halfnormal, hypergeometric

HYPDF   Evaluate the hypergeometric distribution function.

HYPPR   Evaluate the hypergeometric probability function.

L5a1k. Kendall F statistic, Kolmogorov-Smirnov

AKS1DF Evaluate the distribution function of the one-sided Kolmogorov-Smirnov goodness of fit D+ or D test statistic based on continuous data for one sample.

AKS2DF Evaluate the distribution function of the Kolmogorov-Smirnov goodness of fit D test statistic based on continuous data for two samples.

L5a1n. Negative binomial, normal

ANORDF Evaluate the standard normal (Gaussian) distribution function.

L5a1p. Pareto, Poisson

POIDF   Evaluate the Poisson distribution function.

POIPR   Evaluate the Poisson probability function.

L5a1t. t distribution

TDF     Evaluate the Student's t distribution function.

TNDF    Evaluate the noncentral Student's t distribution function.

L5a2.. Inverse cumulative distribution functions, sparsity functions

GCIN    Evaluate the inverse of a general continuous cumulative distribution function given ordinates of the density.

L5a2b. Beta, binomial

BETIN   Evaluate the inverse of the beta distribution function.

L5a2c. Cauchy, chi-squared

CHIIN   Evaluate the inverse of the chi-squared distribution function.

L5a2f. F distribution

FIN     Evaluate the inverse of the F distribution function.

L5a2n. Negative binomial, normal, normal scores

ANORIN Evaluate the inverse of the standard normal (Gaussian) distribution function.

L5a2t. t distribution

TIN     Evaluate the inverse of the Student's t distribution function.

L5b.... Multivariate

L5b1.. Cumulative distribution functions, probability density functions

L5b1n. Normal

BNRDF   Evaluate the bivariate normal distribution function.

N....... DATA HANDLING

N1..... Input, output

IFNAN   Check if a value is NaN (not a number).

N4..... Storage management (e.g., stacks, heaps, trees)

IWKCIN Initialize bookkeeping locations describing the character workspace stack.

IWKIN   Initialize bookkeeping locations describing the workspace stack.

R....... SERVICE ROUTINES

R1..... Machine-dependent constants

AMACH   Retrieve single-precision machine constants.

DMACH   Retrieve double precision machine constants.

IFNAN   Check if a value is NaN (not a number).

IMACH   Retrieve integer machine constants.

UMACH   Set or retrieve input or output device unit numbers.

R3..... Error handling

ERSET   Set error handler default print and stop actions.

IERCD   Retrieve the code for an informational error.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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