29 Numerics library [numerics]

29.7 Mathematical functions for floating-point types [c.math]

29.7.6 Mathematical special functions [sf.cmath]

29.7.6.1 General [sf.cmath.general]

If any argument value to any of the functions specified in [sf.cmath] is a NaN (Not a Number), the function shall return a NaN but it shall not report a domain error.

Otherwise, the function shall report a domain error for just those argument values for which:

(1.1)
the function description's Returns: element explicitly specifies a domain and those argument values fall outside the specified domain, or
(1.2)
the corresponding mathematical function value has a nonzero imaginary component, or
(1.3)
the corresponding mathematical function is not mathematically defined.259

Unless otherwise specified, each function is defined for all finite values, for negative infinity, and for positive infinity.

259)259)

A mathematical function is mathematically defined for a given set of argument values (a) if it is explicitly defined for that set of argument values, or (b) if its limiting value exists and does not depend on the direction of approach.

29.7.6.2 Associated Laguerre polynomials [sf.cmath.assoc.laguerre]

🔗

floating-point-type assoc_laguerre(unsigned n, unsigned m, floating-point-type x);
float        assoc_laguerref(unsigned n, unsigned m, float x);
long double  assoc_laguerrel(unsigned n, unsigned m, long double x);

Effects: These functions compute the associated Laguerre polynomials of their respective arguments n, m, and x.

Returns:

L_{n}^{m} (x)

, where

L_{n}^{m}

is given by Formula 29.22,

L_{n + m}

is given by Formula 29.37, n is n, m is m, and x is x.

\begin{matrix} L_{n}^{m} (x) = (- 1)^{m} \frac{d^{m}}{d x^{m}} L_{n + m} (x), for x \geq 0 \\ (29.22) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128 or if m >= 128.

29.7.6.3 Associated Legendre functions [sf.cmath.assoc.legendre]

🔗

floating-point-type assoc_legendre(unsigned l, unsigned m, floating-point-type x);
float        assoc_legendref(unsigned l, unsigned m, float x);
long double  assoc_legendrel(unsigned l, unsigned m, long double x);

Effects: These functions compute the associated Legendre functions of their respective arguments l, m, and x.

Returns:

P_{ℓ}^{m} (x)

, where

P_{ℓ}^{m}

is given by Formula 29.23,

P_{ℓ}

is given by Formula 29.38, ℓ is l, m is m, and x is x.

\begin{matrix} P_{ℓ}^{m} (x) = (1 - x^{2})^{m / 2} \frac{d^{m}}{d x^{m}} P_{ℓ} (x), for | x | \leq 1 \\ (29.23) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.

29.7.6.4 Beta function [sf.cmath.beta]

🔗

floating-point-type beta(floating-point-type x, floating-point-type y);
float        betaf(float x, float y);
long double  betal(long double x, long double y);

Effects: These functions compute the beta function of their respective arguments x and y.

Returns:

B (x, y)

, where B is given by Formula 29.24, x is x and y is y.

\begin{matrix} B (x, y) = \frac{Γ (x) Γ (y)}{Γ (x + y)}, for x > 0, y > 0 \\ (29.24) \end{matrix}

29.7.6.5 Complete elliptic integral of the first kind [sf.cmath.comp.ellint.1]

🔗

floating-point-type comp_ellint_1(floating-point-type k);
float        comp_ellint_1f(float k);
long double  comp_ellint_1l(long double k);

Effects: These functions compute the complete elliptic integral of the first kind of their respective arguments k.

Returns: K(k), where K is given by Formula 29.25 and k is k.

\begin{matrix} K (k) = F (k, π / 2), for | k | \leq 1 \\ (29.25) \end{matrix}

29.7.6.6 Complete elliptic integral of the second kind [sf.cmath.comp.ellint.2]

🔗

floating-point-type comp_ellint_2(floating-point-type k);
float        comp_ellint_2f(float k);
long double  comp_ellint_2l(long double k);

Effects: These functions compute the complete elliptic integral of the second kind of their respective arguments k.

Returns: E(k), where E is given by Formula 29.26 and k is k.

\begin{matrix} E (k) = E (k, π / 2), for | k | \leq 1 \\ (29.26) \end{matrix}

29.7.6.7 Complete elliptic integral of the third kind [sf.cmath.comp.ellint.3]

🔗

floating-point-type comp_ellint_3(floating-point-type k, floating-point-type nu);
float        comp_ellint_3f(float k, float nu);
long double  comp_ellint_3l(long double k, long double nu);

Effects: These functions compute the complete elliptic integral of the third kind of their respective arguments k and nu.

Returns:

Π (ν, k)

, where Π is given by Formula 29.27, k is k, and ν is nu.

\begin{matrix} Π (ν, k) = Π (ν, k, π / 2), for | k | \leq 1 \\ (29.27) \end{matrix}

29.7.6.8 Regular modified cylindrical Bessel functions [sf.cmath.cyl.bessel.i]

🔗

floating-point-type cyl_bessel_i(floating-point-type nu, floating-point-type x);
float        cyl_bessel_if(float nu, float x);
long double  cyl_bessel_il(long double nu, long double x);

Effects: These functions compute the regular modified cylindrical Bessel functions of their respective arguments nu and x.

Returns:

I_{ν} (x)

, where

I_{ν}

is given by Formula 29.28, ν is nu, and x is x.

\begin{matrix} I_{ν} (x) = i^{- ν} J_{ν} (i x) = \infty \sum k = 0 \frac{(x / 2)^{ν + 2 k}}{k! Γ (ν + k + 1)}, for x \geq 0 \\ (29.28) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

29.7.6.9 Cylindrical Bessel functions of the first kind [sf.cmath.cyl.bessel.j]

🔗

floating-point-type cyl_bessel_j(floating-point-type nu, floating-point-type x);
float        cyl_bessel_jf(float nu, float x);
long double  cyl_bessel_jl(long double nu, long double x);

Effects: These functions compute the cylindrical Bessel functions of the first kind of their respective arguments nu and x.

Returns:

J_{ν} (x)

, where

J_{ν}

is given by Formula 29.29, ν is nu, and x is x.

\begin{matrix} J_{ν} (x) = \infty \sum k = 0 \frac{(- 1)^{k} (x / 2)^{ν + 2 k}}{k! Γ (ν + k + 1)}, for x \geq 0 \\ (29.29) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

29.7.6.10 Irregular modified cylindrical Bessel functions [sf.cmath.cyl.bessel.k]

🔗

floating-point-type cyl_bessel_k(floating-point-type nu, floating-point-type x);
float        cyl_bessel_kf(float nu, float x);
long double  cyl_bessel_kl(long double nu, long double x);

Effects: These functions compute the irregular modified cylindrical Bessel functions of their respective arguments nu and x.

Returns:

K_{ν} (x)

, where

K_{ν}

is given by Formula 29.30, ν is nu, and x is x.

\begin{matrix} K_{ν} (x) = (π / 2) i^{ν + 1} (J_{ν} (i x) + i N_{ν} (i x)) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{π}{2} \frac{I_{- ν} (x) - I_{ν} (x)}{sin ν π}, & for x \geq 0 and non-integral ν \frac{π}{2} lim_{μ \to ν} \frac{I_{- μ} (x) - I_{μ} (x)}{sin μ π}, & for x \geq 0 and integral ν \end{matrix} \\ (29.30) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

29.7.6.11 Cylindrical Neumann functions [sf.cmath.cyl.neumann]

🔗

floating-point-type cyl_neumann(floating-point-type nu, floating-point-type x);
float        cyl_neumannf(float nu, float x);
long double  cyl_neumannl(long double nu, long double x);

Effects: These functions compute the cylindrical Neumann functions, also known as the cylindrical Bessel functions of the second kind, of their respective arguments nu and x.

Returns:

N_{ν} (x)

, where

N_{ν}

is given by Formula 29.31, ν is nu, and x is x.

\begin{matrix} N_{ν} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{J_{ν} (x) cos ν π - J_{- ν} (x)}{sin ν π}, & for x \geq 0 and non-integral ν lim_{μ \to ν} \frac{J_{μ} (x) cos μ π - J_{- μ} (x)}{sin μ π}, & for x \geq 0 and integral ν \end{matrix} \\ (29.31) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if nu >= 128.

29.7.6.12 Incomplete elliptic integral of the first kind [sf.cmath.ellint.1]

🔗

floating-point-type ellint_1(floating-point-type k, floating-point-type phi);
float        ellint_1f(float k, float phi);
long double  ellint_1l(long double k, long double phi);

Effects: These functions compute the incomplete elliptic integral of the first kind of their respective arguments k and phi (phi measured in radians).

Returns: F(k, φ), where F is given by Formula 29.32, k is k, and φ is phi.

\begin{matrix} F (k, ϕ) = \int_{0}^{ϕ} \frac{d θ}{\sqrt{1 - k^{2} {sin}^{2} θ}}, for | k | \leq 1 \\ (29.32) \end{matrix}

29.7.6.13 Incomplete elliptic integral of the second kind [sf.cmath.ellint.2]

🔗

floating-point-type ellint_2(floating-point-type k, floating-point-type phi);
float        ellint_2f(float k, float phi);
long double  ellint_2l(long double k, long double phi);

Effects: These functions compute the incomplete elliptic integral of the second kind of their respective arguments k and phi (phi measured in radians).

Returns: E(k, φ), where E is given by Formula 29.33, k is k, and φ is phi.

\begin{matrix} E (k, ϕ) = \int_{0}^{ϕ} \sqrt{1 - k^{2} {sin}^{2} θ} d θ, for | k | \leq 1 \\ (29.33) \end{matrix}

29.7.6.14 Incomplete elliptic integral of the third kind [sf.cmath.ellint.3]

🔗

floating-point-type ellint_3(floating-point-type k, floating-point-type nu,
                             floating-point-type phi);
float        ellint_3f(float k, float nu, float phi);
long double  ellint_3l(long double k, long double nu, long double phi);

Effects: These functions compute the incomplete elliptic integral of the third kind of their respective arguments k, nu, and phi (phi measured in radians).

Returns:

Π (ν, k, ϕ)

, where Π is given by Formula 29.34, ν is nu, k is k, and φ is phi.

\begin{matrix} Π (ν, k, ϕ) = \int_{0}^{ϕ} \frac{d θ}{(1 - ν {sin}^{2} θ) \sqrt{1 - k^{2} {sin}^{2} θ}}, for | k | \leq 1 \\ (29.34) \end{matrix}

29.7.6.15 Exponential integral [sf.cmath.expint]

🔗

floating-point-type expint(floating-point-type x);
float        expintf(float x);
long double  expintl(long double x);

Effects: These functions compute the exponential integral of their respective arguments x.

Returns: Ei(x), where Ei is given by Formula 29.35 and x is x.

\begin{matrix} E i (x) = - \int_{- x}^{\infty} \frac{e^{- t}}{t} d t \\ (29.35) \end{matrix}

29.7.6.16 Hermite polynomials [sf.cmath.hermite]

🔗

floating-point-type hermite(unsigned n, floating-point-type x);
float        hermitef(unsigned n, float x);
long double  hermitel(unsigned n, long double x);

Effects: These functions compute the Hermite polynomials of their respective arguments n and x.

Returns:

H_{n} (x)

, where

H_{n}

is given by Formula 29.36, n is n, and x is x.

\begin{matrix} H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{- x^{2}} \\ (29.36) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

29.7.6.17 Laguerre polynomials [sf.cmath.laguerre]

🔗

floating-point-type laguerre(unsigned n, floating-point-type x);
float        laguerref(unsigned n, float x);
long double  laguerrel(unsigned n, long double x);

Effects: These functions compute the Laguerre polynomials of their respective arguments n and x.

Returns:

L_{n} (x)

, where

L_{n}

is given by Formula 29.37, n is n, and x is x.

\begin{matrix} L_{n} (x) = \frac{e^{x}}{n!} \frac{d^{n}}{d x^{n}} (x^{n} e^{- x}), for x \geq 0 \\ (29.37) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

29.7.6.18 Legendre polynomials [sf.cmath.legendre]

🔗

floating-point-type legendre(unsigned l, floating-point-type x);
float        legendref(unsigned l, float x);
long double  legendrel(unsigned l, long double x);

Effects: These functions compute the Legendre polynomials of their respective arguments l and x.

Returns:

P_{ℓ} (x)

, where

P_{ℓ}

is given by Formula 29.38, l is l, and x is x.

\begin{matrix} P_{ℓ} (x) = \frac{1}{2^{ℓ} ℓ!} \frac{d^{ℓ}}{d x^{ℓ}} (x^{2} - 1)^{ℓ}, for | x | \leq 1 \\ (29.38) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.

29.7.6.19 Riemann zeta function [sf.cmath.riemann.zeta]

🔗

floating-point-type riemann_zeta(floating-point-type x);
float        riemann_zetaf(float x);
long double  riemann_zetal(long double x);

Effects: These functions compute the Riemann zeta function of their respective arguments x.

Returns: ζ(x), where ζ is given by Formula 29.39 and x is x.

\begin{matrix} ζ (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \infty \sum k = 1 k^{- x}, & for x > 1 \frac{1}{1 - 2^{1 - x}} \infty \sum k = 1 (- 1)^{k - 1} k^{- x}, & for 0 \leq x \leq 1 2^{x} π^{x - 1} sin (\frac{π x}{2}) Γ (1 - x) ζ (1 - x), & for x < 0 \end{matrix} \\ (29.39) \end{matrix}

29.7.6.20 Spherical Bessel functions of the first kind [sf.cmath.sph.bessel]

🔗

floating-point-type sph_bessel(unsigned n, floating-point-type x);
float        sph_besself(unsigned n, float x);
long double  sph_bessell(unsigned n, long double x);

Effects: These functions compute the spherical Bessel functions of the first kind of their respective arguments n and x.

Returns:

j_{n} (x)

, where

j_{n}

is given by Formula 29.40, n is n, and x is x.

\begin{matrix} j_{n} (x) = (π / 2 x)^{1 / 2} J_{n + 1 / 2} (x), for x \geq 0 \\ (29.40) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.

29.7.6.21 Spherical associated Legendre functions [sf.cmath.sph.legendre]

🔗

floating-point-type sph_legendre(unsigned l, unsigned m, floating-point-type theta);
float        sph_legendref(unsigned l, unsigned m, float theta);
long double  sph_legendrel(unsigned l, unsigned m, long double theta);

Effects: These functions compute the spherical associated Legendre functions of their respective arguments l, m, and theta (theta measured in radians).

Returns:

Y_{ℓ}^{m} (θ, 0)

, where

Y_{ℓ}^{m}

is given by Formula 29.41, l is l, m is m, and θ is theta.

\begin{matrix} Y_{ℓ}^{m} (θ, ϕ) = (- 1)^{m} {[\frac{(2 ℓ + 1)}{4 π} \frac{(ℓ - m)!}{(ℓ + m)!}]}^{1 / 2} P_{ℓ}^{m} (cos θ) e^{i m ϕ}, for | m | \leq ℓ \\ (29.41) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if l >= 128.

29.7.6.22 Spherical Neumann functions [sf.cmath.sph.neumann]

🔗

floating-point-type sph_neumann(unsigned n, floating-point-type x);
float        sph_neumannf(unsigned n, float x);
long double  sph_neumannl(unsigned n, long double x);

Effects: These functions compute the spherical Neumann functions, also known as the spherical Bessel functions of the second kind, of their respective arguments n and x.

Returns:

n_{n} (x)

, where

n_{n}

is given by Formula 29.42, n is n, and x is x.

\begin{matrix} n_{n} (x) = (π / 2 x)^{1 / 2} N_{n + 1 / 2} (x), for x \geq 0 \\ (29.42) \end{matrix}

Remarks: The effect of calling each of these functions is implementation-defined if n >= 128.