Special Functions

Gamma

Gamma.gamma(x) computes the Gamma function, \(\Gamma(x)\) (see MathWorld, DLMF). The accuracy of the Hipparchus implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval	Values tested	Average error	Standard deviation	Maximum error
\(-5 < x < -4\)	x[i] = i / 1024, i = -5119, ..., -4097	0.49 ulps	0.57 ulps	3.0 ulps
\(-4 < x < -3\)	x[i] = i / 1024, i = -4095, ..., -3073	0.36 ulps	0.51 ulps	2.0 ulps
\(-3 < x < -2\)	x[i] = i / 1024, i = -3071, ..., -2049	0.41 ulps	0.53 ulps	2.0 ulps
\(-2 < x < -1\)	x[i] = i / 1024, i = -2047, ..., -1025	0.37 ulps	0.50 ulps	2.0 ulps
\(-1 < x < 0\)	x[i] = i / 1024, i = -1023, ..., -1	0.46 ulps	0.54 ulps	2.0 ulps
\(0 < x ≤ 8\)	x[i] = i / 1024, i = 1, ..., 8192	0.33 ulps	0.48 ulps	2.0 ulps
\(8 < x ≤ 141\)	x[i] = i / 64, i = 513, ..., 9024	1.32 ulps	1.19 ulps	7.0 ulps

Log Gamma

Gamma.logGamma(x) computes the natural logarithm of the Gamma function, \(\log \Gamma(x)\), for \(x > 0\) (see MathWorld, DLMF). The accuracy of the Hipparchus implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval	Values tested	Average error	Standard deviation	Maximum error
\(0 < x \le 8\)	x[i] = i / 1024, i = 1, ..., 8192	0.32 ulps	0.50 ulps	4.0 ulps
\(8 < x \le 1024\)	x[i] = i / 8, i = 65, ..., 8192	0.43 ulps	0.53 ulps	3.0 ulps
\(1024 < x \le 8192\)	x[i], i = 1025, ..., 8192	0.53 ulps	0.56 ulps	3.0 ulps
\(8933.439345993791 \le x \le 1.75555970201398e+305\)	x[i] = 2**(i / 8), i = 105, ..., 8112	0.35 ulps	0.49 ulps	2.0 ulps

Regularized Gamma

Gamma.regularizedGammaP(a, x) computes the value of the regularized Gamma function, P(a, x) (see MathWorld).

Log Beta

Beta.logBeta(a, b) computes the value of the natural logarithm of the Beta function, log B(a, b). (see MathWorld, DLMF). The accuracy of the Hipparchus implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval	Values tested	Average error	Standard deviation	Maximum error
\(0 < x \le 8\) \(0 < y \le 8\)	x[i] = i / 32, i = 1, ..., 256 y[j] = j / 32, j = 1, ..., 256	1.80 ulps	81.08 ulps	14031.0 ulps
\(0 < x \le 8\) \(8 < y \le 16\)	x[i] = i / 32, i = 1, ..., 256 y[j] = j / 32, j = 257, ..., 512	0.50 ulps	3.64 ulps	694.0 ulps
\(0 < x \le 8\) \(16 < y \le 256\)	x[i] = i / 32, i = 1, ..., 256 y[j] = j, j = 17, ..., 256	1.04 ulps	139.32 ulps	34509.0 ulps
\(8 < x \le 16\) \(8 < y \le 16\)	x[i] = i / 32, i = 257, ..., 512 y[j] = j / 32, j = 257, ..., 512	0.35 ulps	0.48 ulps	2.0 ulps
\(8 < x \le 16\) \(16 < y \le 256\)	x[i] = i / 32, i = 257, ..., 512 y[j] = j, j = 17, ..., 256	0.31 ulps	0.47 ulps	2.0 ulps
\(16 < x \le 256\) \(16 < y \le 256\)	x[i] = i, i = 17, ..., 256 y[j] = j, j = 17, ..., 256	0.35 ulps	0.49 ulps	2.0 ulps

Regularized Beta

(see MathWorld)

Elliptic functions and integrals

Notations in the domain of elliptic functions and integrals is often confusing and inconsistent across text books. Hipparchus implementation uses the parameter m to define both Jacobi elliptic functions and Legendre elliptic integrals and uses the nome q to define Jacobi theta functions. The elliptic modulus k (which is the square of parameter m) is not used at all in Hipparchus. All these parameters are linked together.

See in MathWorld elliptic integrals of the first kind, elliptic integrals of the second kind, elliptic integrals of the third kind, Jacobi elliptic functions and Jacobi theta functions.

JacobiEllipticBuilder.build(m) builds a JacobiElliptic (or FieldJacobiElliptic) implementation for the parameter m that computes the twelve elliptic functions \(sn(u|m)\), \(cn(u|m)\), \(dn(u|m)\), \(cs(u|m)\), \(ds(u|m)\), \(ns(u|m)\), \(dc(u|m)\), \(nc(u|m)\), \(sc(u|m)\), \(nd(u|m)\), \(sd(u|m)\), and \(cd(u|m)\). The functions are computed as copolar triplets as when one function is needed in an expression, the two other are often also needed and the (very fast) arithmetic-geometric mean algorithm gives all three values at once.

JacobiTheta (and FieldJacobiTheta) computes the four Jacobi theta functions \(\theta_1(z|\tau)\), \(\theta_2(z|\tau)\), \(\theta_3(z|\tau)\), and \(\theta_4(z|\tau)\). The half-period ratio \(\tau\) is linked to the nome q: \(q = e^{i\pi\tau}\). Here again, the four functions are computed at once and a quadruplet is returned.

CarlsonEllipticIntegrals is a utility class that computes the following integrals in Carlson symmetric form, for both primitive double, CalculusFieldElement, Complex and FieldComplex:

Name	Definition
\(R_F(x,y,z)\)	\(\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)}\)
\(R_J(x,y,z,p)\)	\(\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+p)}\)
\(R_G(x,y,z)\)	\(\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)}\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \)
\(R_D(x,y,z)\)	\(R_J(x,y,z,z)\)
\(R_C(x,y)\)	\(R_F(x,y,y)\)

where \(s(t) = \sqrt{t+x}\sqrt{t+y}\sqrt{t+z}\).

LegendreEllipticIntegrals is a utility class that computes the following integrals, for both primitive double, CalculusFieldElement, Complex and FieldComplex. (the implementation uses CarlsonEllipticIntegrals internally):

Name	Type	Definition
\(K(m)\)	complete	\(\int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}}\)
\(K'(m)\)	complete	\(\int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-(1-m) \sin^2\theta}}\)
\(E(m) \)	complete	\(\int_0^{\frac{\pi}{2}} \sqrt{1-m \sin^2\theta} d\theta\)
\(D(m) \)	complete	\(\frac{K(m) - E(m)}{m}\)
\(\Pi(n, m)\)	complete	\(\int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)}\)
\(F(\phi, m)\)	incomplete	\(\int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}}\)
\(E(\phi, m)\)	incomplete	\(\int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta\)
\(D(\phi, m)\)	incomplete	\(\frac{K(\phi, m) - E(\phi, m)}{m}\)
\(\Pi(n, \phi, m)\)	incomplete	\(\int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)}\)