26 Numerics library [numerics]

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

26.8.6 Mathematical special functions [sf.cmath]

26.8.6.1 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.260

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

260)

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.

⮥

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

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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);

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

Returns:

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

where n is n, m is m, and x is x.

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

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

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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);

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

Returns:

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

where l is l, m is m, and x is x.

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

26.8.6.4 Beta function [sf.cmath.beta]

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double       beta(double x, double 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) = \frac{Γ (x) Γ (y)}{Γ (x + y)}, for x > 0, y > 0,

where x is x and y is y.

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

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double       comp_ellint_1(double 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) = F (k, π / 2), for | k | \leq 1,

where k is k.

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

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double       comp_ellint_2(double 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) = E (k, π / 2), for | k | \leq 1,

where k is k.

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

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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);

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

Returns:

Π (ν, k) = Π (ν, k, π / 2), for | k | \leq 1,

where k is k and ν is nu.

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

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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);

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

Returns:

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,

where ν is nu and x is x.

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

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

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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);

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

Returns:

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

where ν is nu and x is x.

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

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

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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);

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

Returns:

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}

where ν is nu and x is x.

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

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

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double       cyl_neumann(double nu, double 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) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \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}

where ν is nu and x is x.

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

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

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double       ellint_1(double k, double 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) = \int_{0}^{ϕ} \frac{d θ}{\sqrt{1 - k^{2} {sin}^{2} θ}}, for | k | \leq 1,

where k is k and φ is phi.

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

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double       ellint_2(double k, double 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) = \int_{0}^{ϕ} \sqrt{1 - k^{2} {sin}^{2} θ} d θ, for | k | \leq 1,

where k is k and φ is phi.

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

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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);

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

where ν is nu, k is k, and φ is phi.

26.8.6.15 Exponential integral [sf.cmath.expint]

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double       expint(double x);
float        expintf(float x);
long double  expintl(long double x);

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

Returns:

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

where x is x.

26.8.6.16 Hermite polynomials [sf.cmath.hermite]

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double       hermite(unsigned n, double 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) = (- 1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{- x^{2}}

where n is n and x is x.

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

26.8.6.17 Laguerre polynomials [sf.cmath.laguerre]

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double       laguerre(unsigned n, double 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) = \frac{e^{x}}{n!} \frac{d^{n}}{d x^{n}} (x^{n} e^{- x}), for x \geq 0,

where n is n and x is x.

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

26.8.6.18 Legendre polynomials [sf.cmath.legendre]

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double       legendre(unsigned l, double 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) = \frac{1}{2^{ℓ} ℓ!} \frac{d^{ℓ}}{d x^{ℓ}} (x^{2} - 1)^{ℓ}, for | x | \leq 1,

where l is l and x is x.

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

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

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double       riemann_zeta(double 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) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \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}

where x is x.

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

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double       sph_bessel(unsigned n, double 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) = (π / 2 x)^{1 / 2} J_{n + 1 / 2} (x), for x \geq 0,

where n is n and x is x.

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

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

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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);

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} (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,

and l is l, m is m, and θ is theta.

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

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

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double       sph_neumann(unsigned n, double 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) = (π / 2 x)^{1 / 2} N_{n + 1 / 2} (x), for x \geq 0,

where n is n and x is x.

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