Documentation
¶
Overview ¶
Package special implements special mathematical functions in pure Go.
The package collects the classical higher transcendental functions that do not appear in the standard math package, grouped by family:
- Bessel functions of the first and second kind (J0, J1, Jn, Y0, Y1, Yn), the modified Bessel functions (I0, I1, In, K0, K1, Kn), the spherical Bessel functions (SphericalJn, SphericalYn) and the Struve functions (StruveH0, StruveH1).
- Airy functions Ai, Bi and their derivatives.
- Elliptic integrals and related quantities.
- The error function family, including the Fresnel integrals.
- The exponential, sine and cosine integrals.
- The Riemann zeta family (zeta, eta, beta).
- The Lambert W function.
- The gamma family (log-gamma, digamma, incomplete forms).
Every routine is implemented with the Go standard library only, using series expansions for small arguments and asymptotic expansions for large ones so that results remain accurate across the full real line. The functions are deterministic and aim for correctness first: they are validated in the test suite against known closed-form and reference values.
Index ¶
- func Airy(x float64) (ai, aip, bi, bip float64)
- func AiryAi(x float64) float64
- func AiryAiPrime(x float64) float64
- func AiryBi(x float64) float64
- func AiryBiPrime(x float64) float64
- func AiryZeroAi(s int) float64
- func AiryZeroBi(s int) float64
- func BesselI(n int, x float64) float64
- func BesselI0(x float64) float64
- func BesselI0Prime(x float64) float64
- func BesselI0e(x float64) float64
- func BesselI1(x float64) float64
- func BesselI1Prime(x float64) float64
- func BesselI1e(x float64) float64
- func BesselIn(n int, x float64) float64
- func BesselInPrime(n int, x float64) float64
- func BesselIne(n int, x float64) float64
- func BesselInu(nu, x float64) float64
- func BesselJ(n int, x float64) float64
- func BesselJ0(x float64) float64
- func BesselJ0Prime(x float64) float64
- func BesselJ1(x float64) float64
- func BesselJ1Prime(x float64) float64
- func BesselJZero(n, s int) float64
- func BesselJn(n int, x float64) float64
- func BesselJnPrime(n int, x float64) float64
- func BesselJnu(nu, x float64) float64
- func BesselK(n int, x float64) float64
- func BesselK0(x float64) float64
- func BesselK0Prime(x float64) float64
- func BesselK0e(x float64) float64
- func BesselK1(x float64) float64
- func BesselK1Prime(x float64) float64
- func BesselK1e(x float64) float64
- func BesselKn(n int, x float64) float64
- func BesselKnPrime(n int, x float64) float64
- func BesselKne(n int, x float64) float64
- func BesselWronskianIK(x float64) float64
- func BesselWronskianJY(x float64) float64
- func BesselY(n int, x float64) float64
- func BesselY0(x float64) float64
- func BesselY0Prime(x float64) float64
- func BesselY1(x float64) float64
- func BesselY1Prime(x float64) float64
- func BesselYZero(n, s int) float64
- func BesselYn(n int, x float64) float64
- func BesselYnPrime(n int, x float64) float64
- func Beta(a, b float64) float64
- func BetaInc(a, b, x float64) float64
- func Chi(x float64) float64
- func Ci(x float64) float64
- func Dawson(x float64) float64
- func Digamma(x float64) float64
- func DirichletBeta(s float64) float64
- func DirichletEta(s float64) float64
- func E1(x float64) float64
- func Ei(x float64) float64
- func EllipticE(m float64) float64
- func EllipticEInc(phi, m float64) float64
- func EllipticF(phi, m float64) float64
- func EllipticK(m float64) float64
- func EllipticPi(n, m float64) float64
- func EllipticPiInc(n, phi, m float64) float64
- func En(n int, x float64) float64
- func Erf(x float64) float64
- func Erfc(x float64) float64
- func Erfcx(x float64) float64
- func Erfi(x float64) float64
- func Fresnel(x float64) (s, c float64)
- func FresnelC(x float64) float64
- func FresnelS(x float64) float64
- func Gamma(x float64) float64
- func GammaP(a, x float64) float64
- func GammaQ(a, x float64) float64
- func HurwitzZeta(s, a float64) float64
- func HypSinCosIntegral(x float64) (shi, chi float64)
- func IncompleteBeta(a, b, x float64) float64
- func InverseErf(x float64) float64
- func InverseErfc(x float64) float64
- func KelvinBei(x float64) float64
- func KelvinBer(x float64) float64
- func KelvinKei(x float64) float64
- func KelvinKer(x float64) float64
- func LambertW(x float64) float64
- func LambertWm1(x float64) float64
- func Li(x float64) float64
- func Li2(x float64) float64
- func Li3(x float64) float64
- func LogGamma(x float64) float64
- func LowerIncompleteGamma(a, x float64) float64
- func ModifiedSphericalI0(x float64) float64
- func ModifiedSphericalI1(x float64) float64
- func ModifiedSphericalK0(x float64) float64
- func ModifiedSphericalK1(x float64) float64
- func Polygamma(n int, x float64) float64
- func Polylog(s int, x float64) float64
- func ReciprocalGamma(x float64) float64
- func RiccatiBesselChi(n int, x float64) float64
- func RiccatiBesselPsi(n int, x float64) float64
- func Shi(x float64) float64
- func Si(x float64) float64
- func SinCosIntegral(x float64) (si, ci float64)
- func Sinc(x float64) float64
- func SincNorm(x float64) float64
- func SphericalJ0(x float64) float64
- func SphericalJ1(x float64) float64
- func SphericalJn(n int, x float64) float64
- func SphericalJnPrime(n int, x float64) float64
- func SphericalY0(x float64) float64
- func SphericalY1(x float64) float64
- func SphericalYn(n int, x float64) float64
- func SphericalYnPrime(n int, x float64) float64
- func Struve(n int, x float64) float64
- func StruveH0(x float64) float64
- func StruveH1(x float64) float64
- func StruveL0(x float64) float64
- func StruveL1(x float64) float64
- func Trigamma(x float64) float64
- func UpperIncompleteGamma(a, x float64) float64
- func Zeta(s float64) float64
Constants ¶
This section is empty.
Variables ¶
This section is empty.
Functions ¶
func Airy ¶
Airy returns the Airy functions Ai(x), Ai'(x), Bi(x) and Bi'(x) simultaneously. The implementation expresses the Airy functions in terms of modified Bessel functions (for x > 0) and ordinary Bessel functions (for x < 0), with a Maclaurin evaluation near the origin.
func AiryAiPrime ¶
AiryAiPrime returns the derivative of the Airy function of the first kind, Ai'(x).
func AiryBiPrime ¶
AiryBiPrime returns the derivative of the Airy function of the second kind, Bi'(x).
func AiryZeroAi ¶
AiryZeroAi returns an approximation to the s-th real zero (s >= 1) of the Airy function Ai, using the standard asymptotic expansion. The returned value is negative.
func AiryZeroBi ¶
AiryZeroBi returns an approximation to the s-th real zero (s >= 1) of the Airy function Bi, using the standard asymptotic expansion. The returned value is negative.
func BesselI ¶
BesselI returns the modified Bessel function of the first kind of integer order n, Iₙ(x).
func BesselI0 ¶
BesselI0 returns the modified Bessel function of the first kind of order zero, I0(x). The result is even in x and always ≥ 1.
func BesselI0Prime ¶
BesselI0Prime returns the derivative of I0, which equals I1(x).
func BesselI1 ¶
BesselI1 returns the modified Bessel function of the first kind of order one, I1(x). The result is odd in x.
func BesselI1Prime ¶
BesselI1Prime returns the derivative of I1(x).
func BesselIn ¶
BesselIn returns the modified Bessel function of the first kind of integer order n, In(x). Negative orders satisfy In(-n, x) = In(n, x).
func BesselInPrime ¶
BesselInPrime returns the derivative of the modified Bessel function In(x).
func BesselInu ¶
BesselInu returns the modified Bessel function of the first kind of arbitrary real order nu, I_nu(x), for x >= 0. It sums the convergent ascending power series and is accurate for small to moderate arguments.
func BesselJ0Prime ¶
BesselJ0Prime returns the derivative of J0, which equals -J1(x).
func BesselJ1Prime ¶
BesselJ1Prime returns the derivative of J1(x).
func BesselJZero ¶
BesselJZero returns an approximation to the s-th positive zero (s >= 1) of the Bessel function J_n, using McMahon's asymptotic expansion. Accuracy improves rapidly with s.
func BesselJn ¶
BesselJn returns the Bessel function of the first kind of integer order n, Jn(x). It is valid for any integer n (negative orders use the reflection Jn(-n, x) = (-1)^n Jn(n, x)).
func BesselJnPrime ¶
BesselJnPrime returns the derivative of the Bessel function Jn(x).
func BesselJnu ¶
BesselJnu returns the Bessel function of the first kind of arbitrary real order nu, J_nu(x), for x >= 0. It sums the convergent ascending power series and is accurate for small to moderate arguments.
func BesselK ¶
BesselK returns the modified Bessel function of the second kind of integer order n, Kₙ(x), for x > 0.
func BesselK0 ¶
BesselK0 returns the modified Bessel function of the second kind of order zero, K0(x). It is defined for x > 0.
func BesselK0Prime ¶
BesselK0Prime returns the derivative of K0, which equals -K1(x). Defined for x > 0.
func BesselK0e ¶
BesselK0e returns the exponentially scaled K0, namely e^{x} K0(x). Defined for x > 0.
func BesselK1 ¶
BesselK1 returns the modified Bessel function of the second kind of order one, K1(x). It is defined for x > 0.
func BesselK1Prime ¶
BesselK1Prime returns the derivative of K1(x). Defined for x > 0.
func BesselK1e ¶
BesselK1e returns the exponentially scaled K1, namely e^{x} K1(x). Defined for x > 0.
func BesselKn ¶
BesselKn returns the modified Bessel function of the second kind of integer order n, Kn(x). It is defined for x > 0. Negative orders satisfy Kn(-n, x) = Kn(n, x).
func BesselKnPrime ¶
BesselKnPrime returns the derivative of the modified Bessel function Kn(x). Defined for x > 0.
func BesselKne ¶
BesselKne returns the exponentially scaled Kn, namely e^{x} Kn(x). Defined for x > 0.
func BesselWronskianIK ¶
BesselWronskianIK returns the Wronskian I_n(x) K_n'(x) - I_n'(x) K_n(x), which equals -1/x for every order n. Defined for x > 0.
func BesselWronskianJY ¶
BesselWronskianJY returns the Wronskian J_n(x) Y_n'(x) - J_n'(x) Y_n(x), which equals 2/(pi x) for every order n. Defined for x > 0.
func BesselY ¶
BesselY returns the Bessel function of the second kind of integer order n, Yₙ(x), for x > 0.
func BesselY0 ¶
BesselY0 returns the Bessel function of the second kind of order zero, Y0(x). It is defined for x > 0.
func BesselY0Prime ¶
BesselY0Prime returns the derivative of Y0, which equals -Y1(x). Defined for x > 0.
func BesselY1 ¶
BesselY1 returns the Bessel function of the second kind of order one, Y1(x). It is defined for x > 0.
func BesselY1Prime ¶
BesselY1Prime returns the derivative of Y1(x). Defined for x > 0.
func BesselYZero ¶
BesselYZero returns an approximation to the s-th positive zero (s >= 1) of the Bessel function Y_n, using McMahon's asymptotic expansion. Accuracy improves rapidly with s.
func BesselYn ¶
BesselYn returns the Bessel function of the second kind of integer order n, Yn(x). It is defined for x > 0.
func BesselYnPrime ¶
BesselYnPrime returns the derivative of the Bessel function Yn(x). Defined for x > 0.
func BetaInc ¶
BetaInc returns the regularized incomplete beta function,
I_x(a, b) = (1/B(a,b)) ∫₀ˣ t^{a-1}(1-t)^{b-1} dt,
for 0 ≤ x ≤ 1 and a, b > 0.
func Chi ¶
Chi returns the hyperbolic cosine integral,
Chi(x) = γ + ln x + ∫₀ˣ (cosh(t)-1)/t dt, x > 0.
func Dawson ¶
Dawson returns Dawson's integral,
D(x) = e^{-x²} ∫₀ˣ e^{t²} dt.
The implementation uses a rational/series scheme accurate to near machine precision across the whole real line.
func Digamma ¶
Digamma returns the digamma function ψ(x) = Γ'(x)/Γ(x), the logarithmic derivative of the gamma function, for real x that is not a non-positive integer.
func DirichletBeta ¶
DirichletBeta returns the Dirichlet beta function,
β(s) = Σ_{n≥0} (-1)^n (2n+1)^{-s}.
It satisfies β(1) = π/4 and β(2) = Catalan's constant.
func DirichletEta ¶
DirichletEta returns the Dirichlet eta function (alternating zeta),
η(s) = Σ_{n≥1} (-1)^{n-1} n^{-s} = (1 - 2^{1-s}) ζ(s).
func E1 ¶
E1 returns the exponential integral E₁,
E₁(x) = ∫₁^∞ e^{-x t}/t dt = ∫_x^∞ e^{-t}/t dt, x > 0.
For x ≤ 0 the value is defined via analytic continuation as -Ei(-x).
func Ei ¶
Ei returns the exponential integral,
Ei(x) = -∫_{-x}^∞ e^{-t}/t dt (principal value for x > 0).
It is defined for all real x ≠ 0; Ei(0) is -Inf.
func EllipticE ¶
EllipticE returns the complete elliptic integral of the second kind,
E(m) = ∫₀^{π/2} √(1 - m sin²θ) dθ,
with parameter m = k². It is defined for m ≤ 1.
func EllipticEInc ¶
EllipticEInc returns the incomplete elliptic integral of the second kind,
E(φ, m) = ∫₀^φ √(1 - m sin²θ) dθ,
with amplitude φ (radians) and parameter m = k².
func EllipticF ¶
EllipticF returns the incomplete elliptic integral of the first kind,
F(φ, m) = ∫₀^φ dθ / √(1 - m sin²θ),
with amplitude φ (radians) and parameter m = k².
func EllipticK ¶
EllipticK returns the complete elliptic integral of the first kind,
K(m) = ∫₀^{π/2} dθ / √(1 - m sin²θ),
with parameter m = k². It is defined for m < 1.
func EllipticPi ¶
EllipticPi returns the complete elliptic integral of the third kind,
Π(n, m) = ∫₀^{π/2} dθ / ((1 - n sin²θ)√(1 - m sin²θ)),
with characteristic n and parameter m = k².
func EllipticPiInc ¶
EllipticPiInc returns the incomplete elliptic integral of the third kind,
Π(n; φ, m) = ∫₀^φ dθ / ((1 - n sin²θ)√(1 - m sin²θ)).
func En ¶
En returns the generalized exponential integral of order n,
Eₙ(x) = ∫₁^∞ e^{-x t}/tⁿ dt, x ≥ 0, n ≥ 0.
E₀(x) = e^{-x}/x and E₁ coincides with the two-argument E1.
func Erf ¶
Erf returns the error function of x,
erf(x) = (2/√π) ∫₀ˣ e^{-t²} dt.
It delegates to the standard library implementation.
func Erfc ¶
Erfc returns the complementary error function of x, erfc(x) = 1 - erf(x). It is accurate even when erf(x) is very close to 1.
func Erfcx ¶
Erfcx returns the scaled complementary error function,
erfcx(x) = e^{x²} · erfc(x).
The scaling removes the exponential decay of erfc for large positive x, avoiding underflow.
func Erfi ¶
Erfi returns the imaginary error function,
erfi(x) = -i·erf(i·x) = (2/√π) ∫₀ˣ e^{t²} dt.
It grows super-exponentially and is related to Dawson's function by erfi(x) = (2/√π)·e^{x²}·D(x).
func Fresnel ¶
Fresnel returns both Fresnel integrals S(x) and C(x) simultaneously,
S(x) = ∫₀ˣ sin(π t²/2) dt, C(x) = ∫₀ˣ cos(π t²/2) dt.
A power series is used for small |x| and an asymptotic auxiliary-function expansion for large |x|.
func Gamma ¶
Gamma returns the gamma function Γ(x); it delegates to the standard library and is provided for API completeness alongside the incomplete forms.
func GammaP ¶
GammaP returns the regularized lower incomplete gamma function,
P(a, x) = (1/Γ(a)) ∫₀ˣ t^{a-1} e^{-t} dt, a > 0, x ≥ 0.
func GammaQ ¶
GammaQ returns the regularized upper incomplete gamma function,
Q(a, x) = 1 - P(a, x) = (1/Γ(a)) ∫ₓ^∞ t^{a-1} e^{-t} dt.
func HurwitzZeta ¶
HurwitzZeta returns the Hurwitz zeta function,
ζ(s, a) = Σ_{n≥0} (n + a)^{-s}, s > 1, a > 0,
extended to s < 1 (s ≠ 1) by Euler–Maclaurin summation.
func HypSinCosIntegral ¶
HypSinCosIntegral returns the hyperbolic sine integral Shi(x) and hyperbolic cosine integral Chi(x) simultaneously. Shi is odd; Chi is evaluated at |x|.
func IncompleteBeta ¶
IncompleteBeta returns the (non-regularized) incomplete beta function B(x; a, b) = ∫₀ˣ t^{a-1}(1-t)^{b-1} dt = I_x(a, b)·B(a, b).
func InverseErf ¶
InverseErf returns the inverse error function, the value y such that erf(y) = x for x in the open interval (-1, 1). It returns ±Inf at ±1 and NaN outside [-1, 1].
func InverseErfc ¶
InverseErfc returns the inverse complementary error function, the value y such that erfc(y) = x for x in the open interval (0, 2). It returns ±Inf at the endpoints and NaN outside [0, 2].
func KelvinBei ¶
KelvinBei returns the Kelvin function bei(x) = Im[J0(x e^{3iπ/4})], summed from its convergent power series.
func KelvinBer ¶
KelvinBer returns the Kelvin function ber(x) = Re[J0(x e^{3iπ/4})], summed from its convergent power series.
func KelvinKei ¶
KelvinKei returns the Kelvin function kei(x) for x > 0, summed from its series with the leading logarithmic terms. It uses
kei(x) = -(ln(x/2)+γ) bei(x) - (π/4) ber(x)
+ Σ_{k≥0} (-1)^k φ(2k+1) / ((2k+1)!)^2 (x/2)^{4k+2}
where φ(n) is the n-th harmonic number.
func KelvinKer ¶
KelvinKer returns the Kelvin function ker(x) for x > 0, summed from its series with the leading logarithmic terms. It uses
ker(x) = -(ln(x/2)+γ) ber(x) + (π/4) bei(x)
+ Σ_{k≥0} (-1)^k φ(2k) / ((2k)!)^2 (x/2)^{4k}
where φ(n) is the n-th harmonic number and φ(0) = 0.
func LambertW ¶
LambertW returns the principal branch W₀(x) of the Lambert W function, the solution w of w·e^w = x for x ≥ -1/e. It returns NaN for x < -1/e.
func LambertWm1 ¶
LambertWm1 returns the secondary real branch W₋₁(x) of the Lambert W function for -1/e ≤ x < 0, the solution w ≤ -1 of w·e^w = x.
func Li ¶
Li returns the logarithmic integral,
li(x) = ∫₀ˣ dt/ln t (principal value for x > 1),
which equals Ei(ln x). It is defined for x > 0, x ≠ 1.
func Li2 ¶
Li2 returns the dilogarithm,
Li₂(x) = Σ_{k≥1} x^k / k² = -∫₀ˣ ln(1-t)/t dt,
for real x ≤ 1. Analytic continuation handles x < -1 and 0 < x ≤ 1.
func Li3 ¶
Li3 returns the trilogarithm,
Li₃(x) = Σ_{k≥1} x^k / k³,
for real x with |x| ≤ 1 (and x < 1). Values outside are returned via the series where it converges.
func LogGamma ¶
LogGamma returns the natural logarithm of the absolute value of the gamma function, ln|Γ(x)|.
func LowerIncompleteGamma ¶
LowerIncompleteGamma returns the (non-regularized) lower incomplete gamma function γ(a, x) = ∫₀ˣ t^{a-1} e^{-t} dt = P(a, x)·Γ(a).
func ModifiedSphericalI0 ¶
ModifiedSphericalI0 returns the modified spherical Bessel function of the first kind of order zero, i0(x) = sinh(x)/x.
func ModifiedSphericalI1 ¶
ModifiedSphericalI1 returns the modified spherical Bessel function of the first kind of order one, i1(x) = (x cosh(x) - sinh(x)) / x^2.
func ModifiedSphericalK0 ¶
ModifiedSphericalK0 returns the modified spherical Bessel function of the second kind of order zero, k0(x) = (pi/2) e^{-x}/x. Defined for x > 0.
func ModifiedSphericalK1 ¶
ModifiedSphericalK1 returns the modified spherical Bessel function of the second kind of order one, k1(x) = (pi/2) e^{-x} (1 + 1/x) / x. Defined for x > 0.
func Polygamma ¶
Polygamma returns the polygamma function ψ⁽ⁿ⁾(x), the n-th derivative of the digamma function, for integer order n ≥ 0 and real x > 0.
ψ⁽ⁿ⁾(x) = (-1)^{n+1} n! Σ_{k≥0} (x+k)^{-(n+1)} for n ≥ 1.
func Polylog ¶
Polylog returns the polylogarithm Liₛ(x) = Σ_{k≥1} x^k / k^s for integer order s and real |x| < 1 (and x = 1 when s > 1, giving ζ(s)). It sums the defining series directly.
func ReciprocalGamma ¶
ReciprocalGamma returns 1/Γ(x), which is entire and vanishes at the non-positive integers.
func RiccatiBesselChi ¶
RiccatiBesselChi returns the Riccati-Bessel function chi_n(x) = -x yn(x).
func RiccatiBesselPsi ¶
RiccatiBesselPsi returns the Riccati-Bessel function psi_n(x) = x jn(x).
func SinCosIntegral ¶
SinCosIntegral returns the sine integral Si(x) and cosine integral Ci(x) simultaneously. For x < 0, Si is odd (Si(-x) = -Si(x)) and Ci(x) is evaluated as Ci(|x|).
func Sinc ¶
Sinc returns the unnormalized cardinal sine, sinc(x) = sin(x)/x, with the removable singularity Sinc(0) = 1 handled exactly.
func SincNorm ¶
SincNorm returns the normalized cardinal sine, sinc(x) = sin(πx)/(πx), with SincNorm(0) = 1.
func SphericalJ0 ¶
SphericalJ0 returns the spherical Bessel function of the first kind of order zero, j0(x) = sin(x)/x, with the removable singularity handled at x = 0.
func SphericalJ1 ¶
SphericalJ1 returns the spherical Bessel function of the first kind of order one, j1(x) = sin(x)/x^2 - cos(x)/x.
func SphericalJn ¶
SphericalJn returns the spherical Bessel function of the first kind of order n, jn(x) = sqrt(pi/(2x)) J_{n+1/2}(x). It uses stable upward recurrence for x ≥ n and downward (Miller) recurrence otherwise.
func SphericalJnPrime ¶
SphericalJnPrime returns the derivative of the spherical Bessel function jn(x), using jn'(x) = j_{n-1}(x) - (n+1)/x * jn(x).
func SphericalY0 ¶
SphericalY0 returns the spherical Bessel function of the second kind of order zero, y0(x) = -cos(x)/x. It is defined for x > 0.
func SphericalY1 ¶
SphericalY1 returns the spherical Bessel function of the second kind of order one, y1(x) = -cos(x)/x^2 - sin(x)/x. It is defined for x > 0.
func SphericalYn ¶
SphericalYn returns the spherical Bessel function of the second kind of order n, yn(x). It is defined for x > 0 and uses stable upward recurrence.
func SphericalYnPrime ¶
SphericalYnPrime returns the derivative of the spherical Bessel function yn(x), using yn'(x) = y_{n-1}(x) - (n+1)/x * yn(x). Defined for x > 0.
func Struve ¶
Struve returns the Struve function Hₙ(x) of integer order n by its power series,
Hₙ(x) = Σ_{m≥0} (-1)^m /(Γ(m+3/2) Γ(m+n+3/2)) (x/2)^{2m+n+1}.
func StruveH0 ¶
StruveH0 returns the Struve function of order zero, H0(x). It is odd in x and uses a power series for small |x| and the relation to Y0 for large |x|.
func StruveH1 ¶
StruveH1 returns the Struve function of order one, H1(x). It is even in x and uses a power series for small |x| and the relation to Y1 for large |x|.
func StruveL0 ¶
StruveL0 returns the modified Struve function of order zero, L0(x). It is odd in x and uses the all-positive-term power series for moderate |x|.
func StruveL1 ¶
StruveL1 returns the modified Struve function of order one, L1(x). It is even in x and uses the all-positive-term power series for moderate |x|.
func Trigamma ¶
Trigamma returns the trigamma function ψ⁽¹⁾(x), the first derivative of the digamma function, for real x that is not a non-positive integer.
func UpperIncompleteGamma ¶
UpperIncompleteGamma returns the (non-regularized) upper incomplete gamma function Γ(a, x) = ∫ₓ^∞ t^{a-1} e^{-t} dt = Q(a, x)·Γ(a).
Types ¶
This section is empty.
Source Files