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<html>
<head>
<title>
SPECIAL_FUNCTIONS - Evaluation of Special Functions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPECIAL_FUNCTIONS <br> Evaluation of Special Functions
</h1>
<hr>
<p>
<b>SPECIAL_FUNCTIONS</b>
is a FORTRAN90 library which
computes the value of various special functions,
by Shanjie Zhang, Jianming Jin.
</p>
<p>
Jianming Jin makes the text of the original FORTRAN77 source code available at
<a href = "http://in.ece.illinois.edu/routines/routines.html">
http://in.ece.illinois.edu/routines/routines.html<a>.
<h3 align = "center">
Licensing:
</h3>
<p>
The FORTRAN77 source code of this library is copyrighted by
Shanjie Zhang and Jianming Jin. However, they give permission to
incorporate routines from this library into a user program
provided that the copyright is acknowledged.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPECIAL_FUNCTIONS</b> is available in
<a href = "../../f77_src/special_functions/special_functions.html">a FORTRAN77 version</a> and
<a href = "../../f_src/special_functions/special_functions.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/cordic/cordic.html">
CORDIC</a>,
a FORTRAN90 library which
uses the CORDIC method to compute certain elementary functions.
</p>
<p>
<a href = "../../f_src/fn/fn.html">
FN</a>,
a FORTRAN90 library which
evaluates elementary and special functions,
by Wayne Fullerton.
</p>
<p>
<a href = "../../f_src/polpak/polpak.html">
POLPAK</a>,
a FORTRAN90 library which
evaluates certain mathematical functions, especially some
recursive polynomial families.
</p>
<p>
<a href = "../../f_src/slatec/slatec.html">
SLATEC</a>,
a FORTRAN90 library which
evaluates many special functions.
</p>
<p>
<a href = "../../f_src/specfun/specfun.html">
SPECFUN</a>,
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;
</p>
<p>
<a href = "../../f_src/test_values/test_values.html">
TEST_VALUES</a>,
a FORTRAN90 library which
contains a few test values of many functions.
</p>
<p>
<a href = "../../f77_src/toms644/toms644.html">
TOMS644</a>,
a FORTRAN77 library which
evaluates the Bessel I, J, K, Y functions, the Airy functions Ai and Bi,
and the Hankel function, for complex argument and real order.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Shanjie Zhang, Jianming Jin,<br>
Computation of Special Functions,<br>
Wiley, 1996,<br>
ISBN: 0-471-11963-6,<br>
LC: QA351.C45.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "special_functions.f90">special_functions.f90</a>, the source code.
</li>
<li>
<a href = "special_functions.sh">special_functions.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "special_functions_prb.f90">special_functions_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "special_functions_prb.sh">special_functions_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "special_functions_prb_output.txt">special_functions_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>AIRYA</b> computes Airy functions and their derivatives.
</li>
<li>
<b>AIRYB</b> computes Airy functions and their derivatives.
</li>
<li>
<b>AIRYZO</b> computes the first NT zeros of Ai(x) and Ai'(x).
</li>
<li>
<b>AJYIK</b> computes Bessel functions Jv(x), Yv(x), Iv(x), Kv(x).
</li>
<li>
<b>ASWFA:</b> prolate and oblate spheroidal angular functions of the first kind.
</li>
<li>
<b>ASWFB:</b> prolate and oblate spheroidal angular functions of the first kind.
</li>
<li>
<b>BERNOA</b> computes the Bernoulli number Bn.
</li>
<li>
<b>BERNOB</b> computes the Bernoulli number Bn.
</li>
<li>
<b>BETA</b> computes the Beta function B(p,q).
</li>
<li>
<b>BJNDD</b> computes Bessel functions Jn(x) and first and second derivatives.
</li>
<li>
<b>CBK</b> computes coefficients for oblate radial functions with small argument.
</li>
<li>
<b>CCHG</b> computes the confluent hypergeometric function.
</li>
<li>
<b>CERF</b> computes the error function and derivative for a complex argument.
</li>
<li>
<b>CERROR</b> computes the error function for a complex argument.
</li>
<li>
<b>CERZO</b> evaluates the complex zeros of the error function.
</li>
<li>
<b>CFC</b> computes the complex Fresnel integral C(z) and C'(z).
</li>
<li>
<b>CFS</b> computes the complex Fresnel integral S(z) and S'(z).
</li>
<li>
<b>CGAMA</b> computes the Gamma function for complex argument.
</li>
<li>
<b>CH12N</b> computes Hankel functions of first and second kinds, complex argument.
</li>
<li>
<b>CHGM</b> computes the confluent hypergeometric function M(a,b,x).
</li>
<li>
<b>CHGU</b> computes the confluent hypergeometric function U(a,b,x).
</li>
<li>
<b>CHGUBI:</b> confluent hypergeometric function with integer argument B.
</li>
<li>
<b>CHGUIT</b> computes the hypergeometric function using Gauss-Legendre integration.
</li>
<li>
<b>CHGUL:</b> confluent hypergeometric function U(a,b,x) for large argument X.
</li>
<li>
<b>CHGUS:</b> confluent hypergeometric function U(a,b,x) for small argument X.
</li>
<li>
<b>CIK01:</b> modified Bessel I0(z), I1(z), K0(z) and K1(z) for complex argument.
</li>
<li>
<b>CIKLV:</b> modified Bessel functions Iv(z), Kv(z), complex argument, large order.
</li>
<li>
<b>CIKNA:</b> modified Bessel functions In(z), Kn(z), derivatives, complex argument.
</li>
<li>
<b>CIKNB</b> computes complex modified Bessel functions In(z) and Kn(z).
</li>
<li>
<b>CIKVA:</b> modified Bessel functions Iv(z), Kv(z), arbitrary order, complex.
</li>
<li>
<b>CIKVB:</b> modified Bessel functions,Iv(z), Kv(z), arbitrary order, complex.
</li>
<li>
<b>CISIA</b> computes cosine Ci(x) and sine integrals Si(x).
</li>
<li>
<b>CISIB</b> computes cosine and sine integrals.
</li>
<li>
<b>CJK:</b> asymptotic expansion coefficients for Bessel functions of large order.
</li>
<li>
<b>CJY01:</b> complexBessel functions, derivatives, J0(z), J1(z), Y0(z), Y1(z).
</li>
<li>
<b>CJYLV:</b> Bessel functions Jv(z), Yv(z) of complex argument and large order v.
</li>
<li>
<b>CJYNA:</b> Bessel functions and derivatives, Jn(z) and Yn(z) of complex argument.
</li>
<li>
<b>CJYNB:</b> Bessel functions, derivatives, Jn(z) and Yn(z) of complex argument.
</li>
<li>
<b>CJYVA:</b> Bessel functions and derivatives, Jv(z) and Yv(z) of complex argument.
</li>
<li>
<b>CJYVB:</b> Bessel functions and derivatives, Jv(z) and Yv(z) of complex argument.
</li>
<li>
<b>CLPMN:</b> associated Legendre functions and derivatives for complex argument.
</li>
<li>
<b>CLPN</b> computes Legendre functions and derivatives for complex argument.
</li>
<li>
<b>CLQMN:</b> associated Legendre functions and derivatives for complex argument.
</li>
<li>
<b>CLQN:</b> Legendre function Qn(z) and derivative Wn'(z) for complex argument.
</li>
<li>
<b>COMELP</b> computes complete elliptic integrals K(k) and E(k).
</li>
<li>
<b>CPBDN:</b> parabolic cylinder function Dn(z) and Dn'(z) for complex argument.
</li>
<li>
<b>CPDLA</b> computes complex parabolic cylinder function Dn(z) for large argument.
</li>
<li>
<b>CPDSA</b> computes complex parabolic cylinder function Dn(z) for small argument.
</li>
<li>
<b>CPSI</b> computes the psi function for a complex argument.
</li>
<li>
<b>CSPHIK:</b> complex modified spherical Bessel functions and derivatives.
</li>
<li>
<b>CSPHJY:</b> spherical Bessel functions jn(z) and yn(z) for complex argument.
</li>
<li>
<b>CV0</b> computes the initial characteristic value of Mathieu functions.
</li>
<li>
<b>CVA1</b> computes a sequence of characteristic values of Mathieu functions.
</li>
<li>
<b>CVA2</b> computes a specific characteristic value of Mathieu functions.
</li>
<li>
<b>CVF</b> computes F for the characteristic equation of Mathieu functions.
</li>
<li>
<b>CVQL</b> computes the characteristic value of Mathieu functions for q <= 3*m.
</li>
<li>
<b>CVQM</b> computes the characteristic value of Mathieu functions for q <= m*m.
</li>
<li>
<b>CY01</b> computes complex Bessel functions Y0(z) and Y1(z) and derivatives.
</li>
<li>
<b>CYZO</b> computes zeros of complex Bessel functions Y0(z) and Y1(z) and Y1'(z).
</li>
<li>
<b>DVLA</b> computes parabolic cylinder functions Dv(x) for large argument.
</li>
<li>
<b>DVSA</b> computes parabolic cylinder functions Dv(x) for small argument.
</li>
<li>
<b>E1XA</b> computes the exponential integral E1(x).
</li>
<li>
<b>E1XB</b> computes the exponential integral E1(x).
</li>
<li>
<b>E1Z</b> computes the complex exponential integral E1(z).
</li>
<li>
<b>EIX</b> computes the exponential integral Ei(x).
</li>
<li>
<b>ELIT:</b> complete and incomplete elliptic integrals F(k,phi) and E(k,phi).
</li>
<li>
<b>ELIT3</b> computes the elliptic integral of the third kind.
</li>
<li>
<b>ENVJ</b> is a utility function used by MSTA1 and MSTA2.
</li>
<li>
<b>ENXA</b> computes the exponential integral En(x).
</li>
<li>
<b>ENXB</b> computes the exponential integral En(x).
</li>
<li>
<b>ERROR</b> evaluates the error function.
</li>
<li>
<b>EULERA</b> computes the Euler number En.
</li>
<li>
<b>EULERB</b> computes the Euler number En.
</li>
<li>
<b>FCOEF:</b> expansion coefficients for Mathieu and modified Mathieu functions.
</li>
<li>
<b>FCS</b> computes Fresnel integrals C(x) and S(x).
</li>
<li>
<b>FCSZO</b> computes complex zeros of Fresnel integrals C(x) or S(x).
</li>
<li>
<b>FFK</b> computes modified Fresnel integrals F+/-(x) and K+/-(x).
</li>
<li>
<b>GAIH</b> computes the GammaH function.
</li>
<li>
<b>GAM0</b> computes the Gamma function for the LAMV function.
</li>
<li>
<b>GAMMA</b> evaluates the Gamma function.
</li>
<li>
<b>GMN</b> computes quantities for oblate radial functions with small argument.
</li>
<li>
<b>HERZO</b> computes the zeros the Hermite polynomial Hn(x).
</li>
<li>
<b>HYGFX</b> evaluates the hypergeometric function F(A,B,C,X).
</li>
<li>
<b>HYGFZ</b> computes the hypergeometric function F(a,b,c,x) for complex argument.
</li>
<li>
<b>IK01A</b> compute Bessel function I0(x), I1(x), K0(x), and K1(x).
</li>
<li>
<b>IK01B:</b> Bessel functions I0(x), I1(x), K0(x), and K1(x) and derivatives.
</li>
<li>
<b>IKNA</b> compute Bessel function In(x) and Kn(x), and derivatives.
</li>
<li>
<b>IKNB</b> compute Bessel function In(x) and Kn(x).
</li>
<li>
<b>IKV</b> compute modified Bessel function Iv(x) and Kv(x) and their derivatives.
</li>
<li>
<b>INCOB</b> computes the incomplete beta function Ix(a,b).
</li>
<li>
<b>INCOG</b> computes the incomplete gamma function r(a,x), ,(a,x), P(a,x).
</li>
<li>
<b>ITAIRY</b> computes the integrals of Airy functions.
</li>
<li>
<b>ITIKA</b> computes the integral of the modified Bessel functions I0(t) and K0(t).
</li>
<li>
<b>ITIKB</b> computes the integral of the Bessel functions I0(t) and K0(t).
</li>
<li>
<b>ITJYA</b> computes integrals of Bessel functions J0(t) and Y0(t).
</li>
<li>
<b>ITJYB</b> computes integrals of Bessel functions J0(t) and Y0(t).
</li>
<li>
<b>ITSH0</b> integrates the Struve function H0(t) from 0 to x.
</li>
<li>
<b>ITSL0</b> integrates the Struve function L0(t) from 0 to x.
</li>
<li>
<b>ITTH0</b> integrates H0(t)/t from x to oo.
</li>
<li>
<b>ITTIKA</b> integrates (I0(t)-1)/t from 0 to x, K0(t)/t from x to infinity.
</li>
<li>
<b>ITTIKB</b> integrates (I0(t)-1)/t from 0 to x, K0(t)/t from x to infinity.
</li>
<li>
<b>ITTJYA</b> integrates (1-J0(t))/t from 0 to x, and Y0(t)/t from x to infinity.
</li>
<li>
<b>ITTJYB</b> integrates (1-J0(t))/t from 0 to x, and Y0(t)/t from x to infinity.
</li>
<li>
<b>JDZO</b> computes the zeros of Bessel functions Jn(x) and Jn'(x).
</li>
<li>
<b>JELP</b> computes Jacobian elliptic functions SN(u), CN(u), DN(u).
</li>
<li>
<b>JY01A</b> computes Bessel functions J0(x), J1(x), Y0(x), Y1(x) and derivatives.
</li>
<li>
<b>JY01B</b> computes Bessel functions J0(x), J1(x), Y0(x), Y1(x) and derivatives.
</li>
<li>
<b>JYNA</b> computes Bessel functions Jn(x) and Yn(x) and derivatives.
</li>
<li>
<b>JYNB</b> computes Bessel functions Jn(x) and Yn(x) and derivatives.
</li>
<li>
<b>JYNDD:</b> Bessel functions Jn(x) and Yn(x), first and second derivatives.
</li>
<li>
<b>JYV</b> computes Bessel functions Jv(x) and Yv(x) and their derivatives.
</li>
<li>
<b>JYZO</b> computes the zeros of Bessel functions Jn(x), Yn(x) and derivatives.
</li>
<li>
<b>KLVNA:</b> Kelvin functions ber(x), bei(x), ker(x), and kei(x), and derivatives.
</li>
<li>
<b>KLVNB:</b> Kelvin functions ber(x), bei(x), ker(x), and kei(x), and derivatives.
</li>
<li>
<b>KLVNZO</b> computes zeros of the Kelvin functions.
</li>
<li>
<b>KMN:</b> expansion coefficients of prolate or oblate spheroidal functions.
</li>
<li>
<b>LAGZO</b> computes zeros of the Laguerre polynomial, and integration weights.
</li>
<li>
<b>LAMN</b> computes lambda functions and derivatives.
</li>
<li>
<b>LAMV</b> computes lambda functions and derivatives of arbitrary order.
</li>
<li>
<b>LEGZO</b> computes the zeros of Legendre polynomials, and integration weights.
</li>
<li>
<b>LGAMA</b> computes the gamma function or its logarithm.
</li>
<li>
<b>LPMN</b> computes associated Legendre functions Pmn(X) and derivatives P'mn(x).
</li>
<li>
<b>LPMNS</b> computes associated Legendre functions Pmn(X) and derivatives P'mn(x).
</li>
<li>
<b>LPMV</b> computes associated Legendre functions Pmv(X) with arbitrary degree.
</li>
<li>
<b>LPN</b> computes Legendre polynomials Pn(x) and derivatives Pn'(x).
</li>
<li>
<b>LPNI</b> computes Legendre polynomials Pn(x), derivatives, and integrals.
</li>
<li>
<b>LQMN</b> computes associated Legendre functions Qmn(x) and derivatives.
</li>
<li>
<b>LQMNS</b> computes associated Legendre functions Qmn(x) and derivatives Qmn'(x).
</li>
<li>
<b>LQNA</b> computes Legendre function Qn(x) and derivatives Qn'(x).
</li>
<li>
<b>LQNB</b> computes Legendre function Qn(x) and derivatives Qn'(x).
</li>
<li>
<b>MSTA1</b> determines a backward recurrence starting point for Jn(x).
</li>
<li>
<b>MSTA2</b> determines a backward recurrence starting point for Jn(x).
</li>
<li>
<b>MTU0</b> computes Mathieu functions CEM(x,q) and SEM(x,q) and derivatives.
</li>
<li>
<b>MTU12</b> computes modified Mathieu functions of the first and second kind.
</li>
<li>
<b>OTHPL</b> computes orthogonal polynomials Tn(x), Un(x), Ln(x) or Hn(x).
</li>
<li>
<b>PBDV</b> computes parabolic cylinder functions Dv(x) and derivatives.
</li>
<li>
<b>PBVV</b> computes parabolic cylinder functions Vv(x) and their derivatives.
</li>
<li>
<b>PBWA</b> computes parabolic cylinder functions W(a,x) and derivatives.
</li>
<li>
<b>PSI</b> computes the PSI function.
</li>
<li>
<b>QSTAR</b> computes Q*mn(-ic) for oblate radial functions with a small argument.
</li>
<li>
<b>RCTJ</b> computes Riccati-Bessel function of the first kind, and derivatives.
</li>
<li>
<b>RCTY</b> computes Riccati-Bessel function of the second kind, and derivatives.
</li>
<li>
<b>REFINE</b> refines an estimate of the characteristic value of Mathieu functions.
</li>
<li>
<b>RMN1</b> computes prolate and oblate spheroidal functions of the first kind.
</li>
<li>
<b>RMN2L:</b> prolate and oblate spheroidal functions, second kind, large CX.
</li>
<li>
<b>RMN2SO:</b> oblate radial functions of the second kind with small argument.
</li>
<li>
<b>RMN2SP:</b> prolate, oblate spheroidal radial functions, kind 2, small argument.
</li>
<li>
<b>RSWFO</b> computes prolate spheroidal radial function of first and second kinds.
</li>
<li>
<b>RSWFP</b> computes prolate spheroidal radial function of first and second kinds.
</li>
<li>
<b>SCKA:</b> expansion coefficients for prolate and oblate spheroidal functions.
</li>
<li>
<b>SCKB:</b> expansion coefficients for prolate and oblate spheroidal functions.
</li>
<li>
<b>SDMN:</b> expansion coefficients for prolate and oblate spheroidal functions.
</li>
<li>
<b>SEGV</b> computes the characteristic values of spheroidal wave functions.
</li>
<li>
<b>SPHI</b> computes spherical Bessel functions in(x) and their derivatives in'(x).
</li>
<li>
<b>SPHJ</b> computes spherical Bessel functions jn(x) and their derivatives.
</li>
<li>
<b>SPHK</b> computes modified spherical Bessel functions kn(x) and derivatives.
</li>
<li>
<b>SPHY</b> computes spherical Bessel functions yn(x) and their derivatives.
</li>
<li>
<b>STVH0</b> computes the Struve function H0(x).
</li>
<li>
<b>STVH1</b> computes the Struve function H1(x).
</li>
<li>
<b>STVHV</b> computes the Struve function Hv(x) with arbitrary order v.
</li>
<li>
<b>STVL0</b> computes the modified Struve function L0(x).
</li>
<li>
<b>STVL1</b> computes the modified Struve function L1(x).
</li>
<li>
<b>STVLV</b> computes the modified Struve function Lv(x) with arbitary order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VVLA</b> computes parabolic cylinder function Vv(x) for large arguments.
</li>
<li>
<b>VVSA</b> computes parabolic cylinder function V(nu,x) for small arguments.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 03 August 2012.
</i>
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