SPECIAL_FUNCTIONS
Evaluation of Special Functions

SPECIAL_FUNCTIONS, a FORTRAN90 library which evaluates special functions, including Airy, Associated Legendre Bessel, Beta, Complete Elliptic Integral, Confluent Hypergeometric, Cosine Integral, Elliptic Integral, Error, Exponential Integral, Fresnel Integral, Gamma, Hankel, Hypergeometric, Incomplete Beta, Incomplete Gamma, Jacobian Elliptic, Kelvin, Lambda, Legendre, Mathieu, Modified Spherical Bessel, Parabolic Cylinder, Psi, Riccati-Bessel, Sine Integral, Spheroidal Wave, Struve, Whittaker, as well as Bernoulli Numbers, Euler Numbers, Hermite Polynomials, Laguerre Polynomials, Legendre Polynomials, by Shanjie Zhang, Jianming Jin.

Jianming Jin makes the text of the original FORTRAN77 source code available at http://in.ece.illinois.edu/routines/routines.html.

Licensing:

SPECIAL_FUNCTIONS is available in a FORTRAN90 version.

Related Data and Programs:

CORDIC, a FORTRAN90 library which uses the CORDIC method to compute certain elementary functions.

FN, a FORTRAN90 library which evaluates elementary and special functions, by Wayne Fullerton.

POLPAK, a FORTRAN90 library which evaluates certain mathematical functions, especially some recursive polynomial families.

SLATEC, a FORTRAN90 library which evaluates many special functions.

SPECFUN, a FORTRAN90 library which computes special functions, including Bessel I, J, K and Y functions, and the Dawson, E1, EI, Erf, Gamma, Psi/Digamma functions, by William Cody and Laura Stoltz;

TEST_VALUES, a FORTRAN90 library which contains a few test values of many functions.

TOMS715, a FORTRAN90 library which evaluates special functions, including the Bessel I, J, K, and Y functions of order 0, of order 1, and of any real order, Dawson's integral, the error function, exponential integrals, the gamma function, the normal distribution function, the psi function. This is a version of ACM TOMS algorithm 715.

Reference:

1. Shanjie Zhang, Jianming Jin,
   Computation of Special Functions,
   Wiley, 1996,
   ISBN: 0-471-11963-6,
   LC: QA351.C45.

Source Code:

• special_functions.f90, the source code.

Examples and Tests:

• special_functions_test.f90, a sample calling program.
• special_functions_test.txt, the output file.

List of Routines:

• AIRYA computes Airy functions and their derivatives.
• AIRYB computes Airy functions and their derivatives.
• AIRYZO computes the first NT zeros of Ai(x) and Ai'(x).
• AJYIK computes Bessel functions Jv(x), Yv(x), Iv(x), Kv(x).
• ASWFA: prolate and oblate spheroidal angular functions of the first kind.
• ASWFB: prolate and oblate spheroidal angular functions of the first kind.
• BERNOA computes the Bernoulli number Bn.
• BERNOB computes the Bernoulli number Bn.
• BETA computes the Beta function B(p,q).
• BJNDD computes Bessel functions Jn(x) and first and second derivatives.
• CBK computes coefficients for oblate radial functions with small argument.
• CCHG computes the confluent hypergeometric function.
• CERF computes the error function and derivative for a complex argument.
• CERROR computes the error function for a complex argument.
• CERZO evaluates the complex zeros of the error function.
• CFC computes the complex Fresnel integral C(z) and C'(z).
• CFS computes the complex Fresnel integral S(z) and S'(z).
• CGAMA computes the Gamma function for complex argument.
• CH12N computes Hankel functions of first and second kinds, complex argument.
• CHGM computes the confluent hypergeometric function M(a,b,x).
• CHGU computes the confluent hypergeometric function U(a,b,x).
• CHGUBI: confluent hypergeometric function with integer argument B.
• CHGUIT computes the hypergeometric function using Gauss-Legendre integration.
• CHGUL: confluent hypergeometric function U(a,b,x) for large argument X.
• CHGUS: confluent hypergeometric function U(a,b,x) for small argument X.
• CIK01: modified Bessel I0(z), I1(z), K0(z) and K1(z) for complex argument.
• CIKLV: modified Bessel functions Iv(z), Kv(z), complex argument, large order.
• CIKNA: modified Bessel functions In(z), Kn(z), derivatives, complex argument.
• CIKNB computes complex modified Bessel functions In(z) and Kn(z).
• CIKVA: modified Bessel functions Iv(z), Kv(z), arbitrary order, complex.
• CIKVB: modified Bessel functions,Iv(z), Kv(z), arbitrary order, complex.
• CISIA computes cosine Ci(x) and sine integrals Si(x).
• CISIB computes cosine and sine integrals.
• CJK: asymptotic expansion coefficients for Bessel functions of large order.
• CJY01: complexBessel functions, derivatives, J0(z), J1(z), Y0(z), Y1(z).
• CJYLV: Bessel functions Jv(z), Yv(z) of complex argument and large order v.
• CJYNA: Bessel functions and derivatives, Jn(z) and Yn(z) of complex argument.
• CJYNB: Bessel functions, derivatives, Jn(z) and Yn(z) of complex argument.
• CJYVA: Bessel functions and derivatives, Jv(z) and Yv(z) of complex argument.
• CJYVB: Bessel functions and derivatives, Jv(z) and Yv(z) of complex argument.
• CLPMN: associated Legendre functions and derivatives for complex argument.
• CLPN computes Legendre functions and derivatives for complex argument.
• CLQMN: associated Legendre functions and derivatives for complex argument.
• CLQN: Legendre function Qn(z) and derivative Wn'(z) for complex argument.
• COMELP computes complete elliptic integrals K(k) and E(k).
• CPBDN: parabolic cylinder function Dn(z) and Dn'(z) for complex argument.
• CPDLA computes complex parabolic cylinder function Dn(z) for large argument.
• CPDSA computes complex parabolic cylinder function Dn(z) for small argument.
• CPSI computes the psi function for a complex argument.
• CSPHIK: complex modified spherical Bessel functions and derivatives.
• CSPHJY: spherical Bessel functions jn(z) and yn(z) for complex argument.
• CV0 computes the initial characteristic value of Mathieu functions.
• CVA1 computes a sequence of characteristic values of Mathieu functions.
• CVA2 computes a specific characteristic value of Mathieu functions.
• CVF computes F for the characteristic equation of Mathieu functions.
• CVQL computes the characteristic value of Mathieu functions for q <= 3*m.
• CVQM computes the characteristic value of Mathieu functions for q <= m*m.
• CY01 computes complex Bessel functions Y0(z) and Y1(z) and derivatives.
• CYZO computes zeros of complex Bessel functions Y0(z) and Y1(z) and Y1'(z).
• DVLA computes parabolic cylinder functions Dv(x) for large argument.
• DVSA computes parabolic cylinder functions Dv(x) for small argument.
• E1XA computes the exponential integral E1(x).
• E1XB computes the exponential integral E1(x).
• E1Z computes the complex exponential integral E1(z).
• EIX computes the exponential integral Ei(x).
• ELIT: complete and incomplete elliptic integrals F(k,phi) and E(k,phi).
• ELIT3 computes the elliptic integral of the third kind.
• ENVJ is a utility function used by MSTA1 and MSTA2.
• ENXA computes the exponential integral En(x).
• ENXB computes the exponential integral En(x).
• ERROR evaluates the error function.
• EULERA computes the Euler number En.
• EULERB computes the Euler number En.
• FCOEF: expansion coefficients for Mathieu and modified Mathieu functions.
• FCS computes Fresnel integrals C(x) and S(x).
• FCSZO computes complex zeros of Fresnel integrals C(x) or S(x).
• FFK computes modified Fresnel integrals F+/-(x) and K+/-(x).
• GAIH computes the GammaH function.
• GAM0 computes the Gamma function for the LAMV function.
• GAMMA evaluates the Gamma function.
• GMN computes quantities for oblate radial functions with small argument.
• HERZO computes the zeros the Hermite polynomial Hn(x).
• HYGFX evaluates the hypergeometric function F(A,B,C,X).
• HYGFZ computes the hypergeometric function F(a,b,c,x) for complex argument.
• IK01A compute Bessel function I0(x), I1(x), K0(x), and K1(x).
• IK01B: Bessel functions I0(x), I1(x), K0(x), and K1(x) and derivatives.
• IKNA compute Bessel function In(x) and Kn(x), and derivatives.
• IKNB compute Bessel function In(x) and Kn(x).
• IKV compute modified Bessel function Iv(x) and Kv(x) and their derivatives.
• INCOB computes the incomplete beta function Ix(a,b).
• INCOG computes the incomplete gamma function r(a,x), ,(a,x), P(a,x).
• ITAIRY computes the integrals of Airy functions.
• ITIKA computes the integral of the modified Bessel functions I0(t) and K0(t).
• ITIKB computes the integral of the Bessel functions I0(t) and K0(t).
• ITJYA computes integrals of Bessel functions J0(t) and Y0(t).
• ITJYB computes integrals of Bessel functions J0(t) and Y0(t).
• ITSH0 integrates the Struve function H0(t) from 0 to x.
• ITSL0 integrates the Struve function L0(t) from 0 to x.
• ITTH0 integrates H0(t)/t from x to oo.
• ITTIKA integrates (I0(t)-1)/t from 0 to x, K0(t)/t from x to infinity.
• ITTIKB integrates (I0(t)-1)/t from 0 to x, K0(t)/t from x to infinity.
• ITTJYA integrates (1-J0(t))/t from 0 to x, and Y0(t)/t from x to infinity.
• ITTJYB integrates (1-J0(t))/t from 0 to x, and Y0(t)/t from x to infinity.
• JDZO computes the zeros of Bessel functions Jn(x) and Jn'(x).
• JELP computes Jacobian elliptic functions SN(u), CN(u), DN(u).
• JY01A computes Bessel functions J0(x), J1(x), Y0(x), Y1(x) and derivatives.
• JY01B computes Bessel functions J0(x), J1(x), Y0(x), Y1(x) and derivatives.
• JYNA computes Bessel functions Jn(x) and Yn(x) and derivatives.
• JYNB computes Bessel functions Jn(x) and Yn(x) and derivatives.
• JYNDD: Bessel functions Jn(x) and Yn(x), first and second derivatives.
• JYV computes Bessel functions Jv(x) and Yv(x) and their derivatives.
• JYZO computes the zeros of Bessel functions Jn(x), Yn(x) and derivatives.
• KLVNA: Kelvin functions ber(x), bei(x), ker(x), and kei(x), and derivatives.
• KLVNB: Kelvin functions ber(x), bei(x), ker(x), and kei(x), and derivatives.
• KLVNZO computes zeros of the Kelvin functions.
• KMN: expansion coefficients of prolate or oblate spheroidal functions.
• LAGZO computes zeros of the Laguerre polynomial, and integration weights.
• LAMN computes lambda functions and derivatives.
• LAMV computes lambda functions and derivatives of arbitrary order.
• LEGZO computes the zeros of Legendre polynomials, and integration weights.
• LGAMA computes the gamma function or its logarithm.
• LPMN computes associated Legendre functions Pmn(X) and derivatives P'mn(x).
• LPMNS computes associated Legendre functions Pmn(X) and derivatives P'mn(x).
• LPMV computes associated Legendre functions Pmv(X) with arbitrary degree.
• LPN computes Legendre polynomials Pn(x) and derivatives Pn'(x).
• LPNI computes Legendre polynomials Pn(x), derivatives, and integrals.
• LQMN computes associated Legendre functions Qmn(x) and derivatives.
• LQMNS computes associated Legendre functions Qmn(x) and derivatives Qmn'(x).
• LQNA computes Legendre function Qn(x) and derivatives Qn'(x).
• LQNB computes Legendre function Qn(x) and derivatives Qn'(x).
• MSTA1 determines a backward recurrence starting point for Jn(x).
• MSTA2 determines a backward recurrence starting point for Jn(x).
• MTU0 computes Mathieu functions CEM(x,q) and SEM(x,q) and derivatives.
• MTU12 computes modified Mathieu functions of the first and second kind.
• OTHPL computes orthogonal polynomials Tn(x), Un(x), Ln(x) or Hn(x).
• PBDV computes parabolic cylinder functions Dv(x) and derivatives.
• PBVV computes parabolic cylinder functions Vv(x) and their derivatives.
• PBWA computes parabolic cylinder functions W(a,x) and derivatives.
• PSI computes the PSI function.
• QSTAR computes Q*mn(-ic) for oblate radial functions with a small argument.
• RCTJ computes Riccati-Bessel function of the first kind, and derivatives.
• RCTY computes Riccati-Bessel function of the second kind, and derivatives.
• REFINE refines an estimate of the characteristic value of Mathieu functions.
• RMN1 computes prolate and oblate spheroidal functions of the first kind.
• RMN2L: prolate and oblate spheroidal functions, second kind, large CX.
• RMN2SO: oblate radial functions of the second kind with small argument.
• RMN2SP: prolate, oblate spheroidal radial functions, kind 2, small argument.
• RSWFO computes prolate spheroidal radial function of first and second kinds.
• RSWFP computes prolate spheroidal radial function of first and second kinds.
• SCKA: expansion coefficients for prolate and oblate spheroidal functions.
• SCKB: expansion coefficients for prolate and oblate spheroidal functions.
• SDMN: expansion coefficients for prolate and oblate spheroidal functions.
• SEGV computes the characteristic values of spheroidal wave functions.
• SPHI computes spherical Bessel functions in(x) and their derivatives in'(x).
• SPHJ computes spherical Bessel functions jn(x) and their derivatives.
• SPHK computes modified spherical Bessel functions kn(x) and derivatives.
• SPHY computes spherical Bessel functions yn(x) and their derivatives.
• STVH0 computes the Struve function H0(x).
• STVH1 computes the Struve function H1(x).
• STVHV computes the Struve function Hv(x) with arbitrary order v.
• STVL0 computes the modified Struve function L0(x).
• STVL1 computes the modified Struve function L1(x).
• STVLV computes the modified Struve function Lv(x) with arbitary order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• VVLA computes parabolic cylinder function Vv(x) for large arguments.
• VVSA computes parabolic cylinder function V(nu,x) for small arguments.

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