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## TR1 C Functions Quick Reference

##### Supported TR1 Functions

    namespace boost{ namespace math{ namespace tr1{ extern "C"{

    // [5.2.1.1] associated Laguerre polynomials:
    double assoc_laguerre(unsigned n, unsigned m, double x);
    float assoc_laguerref(unsigned n, unsigned m, float x);
    long double assoc_laguerrel(unsigned n, unsigned m, long double x);

    // [5.2.1.2] associated Legendre functions:
    double assoc_legendre(unsigned l, unsigned m, double x);
    float assoc_legendref(unsigned l, unsigned m, float x);
    long double assoc_legendrel(unsigned l, unsigned m, long double x);

    // [5.2.1.3] beta function:
    double beta(double x, double y);
    float betaf(float x, float y);
    long double betal(long double x, long double y);

    // [5.2.1.4] (complete) elliptic integral of the first kind:
    double comp_ellint_1(double k);
    float comp_ellint_1f(float k);
    long double comp_ellint_1l(long double k);

    // [5.2.1.5] (complete) elliptic integral of the second kind:
    double comp_ellint_2(double k);
    float comp_ellint_2f(float k);
    long double comp_ellint_2l(long double k);

    // [5.2.1.6] (complete) elliptic integral of the third kind:
    double comp_ellint_3(double k, double nu);
    float comp_ellint_3f(float k, float nu);
    long double comp_ellint_3l(long double k, long double nu);

    // [5.2.1.8] regular modified cylindrical Bessel functions:
    double cyl_bessel_i(double nu, double x);
    float cyl_bessel_if(float nu, float x);
    long double cyl_bessel_il(long double nu, long double x);

    // [5.2.1.9] cylindrical Bessel functions (of the first kind):
    double cyl_bessel_j(double nu, double x);
    float cyl_bessel_jf(float nu, float x);
    long double cyl_bessel_jl(long double nu, long double x);

    // [5.2.1.10] irregular modified cylindrical Bessel functions:
    double cyl_bessel_k(double nu, double x);
    float cyl_bessel_kf(float nu, float x);
    long double cyl_bessel_kl(long double nu, long double x);

    // [5.2.1.11] cylindrical Neumann functions;
    // cylindrical Bessel functions (of the second kind):
    double cyl_neumann(double nu, double x);
    float cyl_neumannf(float nu, float x);
    long double cyl_neumannl(long double nu, long double x);

    // [5.2.1.12] (incomplete) elliptic integral of the first kind:
    double ellint_1(double k, double phi);
    float ellint_1f(float k, float phi);
    long double ellint_1l(long double k, long double phi);

    // [5.2.1.13] (incomplete) elliptic integral of the second kind:
    double ellint_2(double k, double phi);
    float ellint_2f(float k, float phi);
    long double ellint_2l(long double k, long double phi);

    // [5.2.1.14] (incomplete) elliptic integral of the third kind:
    double ellint_3(double k, double nu, double phi);
    float ellint_3f(float k, float nu, float phi);
    long double ellint_3l(long double k, long double nu, long double phi);

    // [5.2.1.15] exponential integral:
    double expint(double x);
    float expintf(float x);
    long double expintl(long double x);

    // [5.2.1.16] Hermite polynomials:
    double hermite(unsigned n, double x);
    float hermitef(unsigned n, float x);
    long double hermitel(unsigned n, long double x);

    // [5.2.1.18] Laguerre polynomials:
    double laguerre(unsigned n, double x);
    float laguerref(unsigned n, float x);
    long double laguerrel(unsigned n, long double x);

    // [5.2.1.19] Legendre polynomials:
    double legendre(unsigned l, double x);
    float legendref(unsigned l, float x);
    long double legendrel(unsigned l, long double x);

    // [5.2.1.20] Riemann zeta function:
    double riemann_zeta(double);
    float riemann_zetaf(float);
    long double riemann_zetal(long double);

    // [5.2.1.21] spherical Bessel functions (of the first kind):
    double sph_bessel(unsigned n, double x);
    float sph_besself(unsigned n, float x);
    long double sph_bessell(unsigned n, long double x);

    // [5.2.1.22] spherical associated Legendre functions:
    double sph_legendre(unsigned l, unsigned m, double theta);
    float sph_legendref(unsigned l, unsigned m, float theta);
    long double sph_legendrel(unsigned l, unsigned m, long double theta);

    // [5.2.1.23] spherical Neumann functions;
    // spherical Bessel functions (of the second kind):
    double sph_neumann(unsigned n, double x);
    float sph_neumannf(unsigned n, float x);
    long double sph_neumannl(unsigned n, long double x);

    }}}} // namespaces

In addition sufficient additional overloads of the `double` versions of the above functions are provided, so that calling the function with any mixture of `float`, `double`, `long double`, or integer arguments is supported, with the return type determined by the result type calculation rules.

For example:

    expintf(2.0f);  // float version, returns float.
    expint(2.0f);   // also calls the float version and returns float.
    expint(2.0);    // double version, returns double.
    expintl(2.0L);  // long double version, returns a long double.
    expint(2.0L);   // also calls the long double version.
    expint(2);      // integer argument is treated as a double, returns double.

##### Quick Reference

    // [5.2.1.1] associated Laguerre polynomials:
    double assoc_laguerre(unsigned n, unsigned m, double x);
    float assoc_laguerref(unsigned n, unsigned m, float x);
    long double assoc_laguerrel(unsigned n, unsigned m, long double x);

The assoc_laguerre functions return:

See also laguerre for the full template (header only) version of this function.

    // [5.2.1.2] associated Legendre functions:
    double assoc_legendre(unsigned l, unsigned m, double x);
    float assoc_legendref(unsigned l, unsigned m, float x);
    long double assoc_legendrel(unsigned l, unsigned m, long double x);

The assoc_legendre functions return:

See also legendre_p for the full template (header only) version of this function.

    // [5.2.1.3] beta function:
    double beta(double x, double y);
    float betaf(float x, float y);
    long double betal(long double x, long double y);

Returns the beta function of x and y:

See also beta for the full template (header only) version of this function.

    // [5.2.1.4] (complete) elliptic integral of the first kind:
    double comp_ellint_1(double k);
    float comp_ellint_1f(float k);
    long double comp_ellint_1l(long double k);

Returns the complete elliptic integral of the first kind of k:

See also ellint_1 for the full template (header only) version of this function.

    // [5.2.1.5] (complete) elliptic integral of the second kind:
    double comp_ellint_2(double k);
    float comp_ellint_2f(float k);
    long double comp_ellint_2l(long double k);

Returns the complete elliptic integral of the second kind of k:

See also ellint_2 for the full template (header only) version of this function.

    // [5.2.1.6] (complete) elliptic integral of the third kind:
    double comp_ellint_3(double k, double nu);
    float comp_ellint_3f(float k, float nu);
    long double comp_ellint_3l(long double k, long double nu);

Returns the complete elliptic integral of the third kind of k and nu:

See also ellint_3 for the full template (header only) version of this function.

    // [5.2.1.8] regular modified cylindrical Bessel functions:
    double cyl_bessel_i(double nu, double x);
    float cyl_bessel_if(float nu, float x);
    long double cyl_bessel_il(long double nu, long double x);

Returns the modified bessel function of the first kind of nu and x:

See also cyl_bessel_i for the full template (header only) version of this function.

    // [5.2.1.9] cylindrical Bessel functions (of the first kind):
    double cyl_bessel_j(double nu, double x);
    float cyl_bessel_jf(float nu, float x);
    long double cyl_bessel_jl(long double nu, long double x);

Returns the bessel function of the first kind of nu and x:

See also cyl_bessel_j for the full template (header only) version of this function.

    // [5.2.1.10] irregular modified cylindrical Bessel functions:
    double cyl_bessel_k(double nu, double x);
    float cyl_bessel_kf(float nu, float x);
    long double cyl_bessel_kl(long double nu, long double x);

Returns the modified bessel function of the second kind of nu and x:

See also cyl_bessel_k for the full template (header only) version of this function.

    // [5.2.1.11] cylindrical Neumann functions;
    // cylindrical Bessel functions (of the second kind):
    double cyl_neumann(double nu, double x);
    float cyl_neumannf(float nu, float x);
    long double cyl_neumannl(long double nu, long double x);

Returns the bessel function of the second kind (Neumann function) of nu and x:

See also cyl_neumann for the full template (header only) version of this function.

    // [5.2.1.12] (incomplete) elliptic integral of the first kind:
    double ellint_1(double k, double phi);
    float ellint_1f(float k, float phi);
    long double ellint_1l(long double k, long double phi);

Returns the incomplete elliptic integral of the first kind of k and phi:

See also ellint_1 for the full template (header only) version of this function.

    // [5.2.1.13] (incomplete) elliptic integral of the second kind:
    double ellint_2(double k, double phi);
    float ellint_2f(float k, float phi);
    long double ellint_2l(long double k, long double phi);

Returns the incomplete elliptic integral of the second kind of k and phi:

See also ellint_2 for the full template (header only) version of this function.

    // [5.2.1.14] (incomplete) elliptic integral of the third kind:
    double ellint_3(double k, double nu, double phi);
    float ellint_3f(float k, float nu, float phi);
    long double ellint_3l(long double k, long double nu, long double phi);

Returns the incomplete elliptic integral of the third kind of k, nu and phi:

See also ellint_3 for the full template (header only) version of this function.

    // [5.2.1.15] exponential integral:
    double expint(double x);
    float expintf(float x);
    long double expintl(long double x);

Returns the exponential integral Ei of x:

See also expint for the full template (header only) version of this function.

    // [5.2.1.16] Hermite polynomials:
    double hermite(unsigned n, double x);
    float hermitef(unsigned n, float x);
    long double hermitel(unsigned n, long double x);

Returns the n'th Hermite polynomial of x:

See also hermite for the full template (header only) version of this function.

    // [5.2.1.18] Laguerre polynomials:
    double laguerre(unsigned n, double x);
    float laguerref(unsigned n, float x);
    long double laguerrel(unsigned n, long double x);

Returns the n'th Laguerre polynomial of x:

See also laguerre for the full template (header only) version of this function.

    // [5.2.1.19] Legendre polynomials:
    double legendre(unsigned l, double x);
    float legendref(unsigned l, float x);
    long double legendrel(unsigned l, long double x);

Returns the l'th Legendre polynomial of x:

See also legendre_p for the full template (header only) version of this function.

    // [5.2.1.20] Riemann zeta function:
    double riemann_zeta(double);
    float riemann_zetaf(float);
    long double riemann_zetal(long double);

Returns the Riemann Zeta function of x:

See also zeta for the full template (header only) version of this function.

    // [5.2.1.21] spherical Bessel functions (of the first kind):
    double sph_bessel(unsigned n, double x);
    float sph_besself(unsigned n, float x);
    long double sph_bessell(unsigned n, long double x);

Returns the spherical Bessel function of the first kind of x jn(x):

See also sph_bessel for the full template (header only) version of this function.

    // [5.2.1.22] spherical associated Legendre functions:
    double sph_legendre(unsigned l, unsigned m, double theta);
    float sph_legendref(unsigned l, unsigned m, float theta);
    long double sph_legendrel(unsigned l, unsigned m, long double theta);

Returns the spherical associated Legendre function of l, m and theta:

See also spherical_harmonic for the full template (header only) version of this function.

    // [5.2.1.23] spherical Neumann functions;
    // spherical Bessel functions (of the second kind):
    double sph_neumann(unsigned n, double x);
    float sph_neumannf(unsigned n, float x);
    long double sph_neumannl(unsigned n, long double x);

Returns the spherical Neumann function of x yn(x):

See also sph_bessel for the full template (header only) version of this function.

##### Currently Unsupported TR1 Functions

    // [5.2.1.7] confluent hypergeometric functions:
    double conf_hyperg(double a, double c, double x);
    float conf_hypergf(float a, float c, float x);
    long double conf_hypergl(long double a, long double c, long double x);

    // [5.2.1.17] hypergeometric functions:
    double hyperg(double a, double b, double c, double x);
    float hypergf(float a, float b, float c, float x);
    long double hypergl(long double a, long double b, long double c,
    long double x);

| Note
These two functions are not implemented as they are not believed to be numerically stable.