Class SpecialFunctions

java.lang.Object

org.episteme.core.mathematics.analysis.special.SpecialFunctions

public class SpecialFunctions
extends Object

Special mathematical functions.

Gamma, Beta, Bessel, Error functions - essential for physics, statistics, engineering.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
SpecialFunctions()

Method Summary

Modifier and Type	Method	Description
static Real	airyAi(Real x)	Airy function Ai(x).
static Real	besselIn(int n, Real x)	Modified Bessel function of the first kind I_n(x).
static Real	besselJ0(Real x)	Bessel function of the first kind Jâ‚€(x).
static Real	besselJn(int n, Real x)	Bessel function of the first kind J_n(x) for integer order.
static Real	besselKn(int n, Real x)	Modified Bessel function of the second kind K_n(x).
static Real	besselY0(Real x)	Bessel function of the second kind Yâ‚€(x).
static Real	besselYn(int n, Real x)	Bessel function of the second kind Y_n(x) for integer order.
static Real	beta(Real x, Real y)	Beta function: B(x, y) = Γ(x)Γ(y) / Γ(x+y)
static Real	erf(Real x)	Error function: erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt
static Real	erfc(Real x)	Complementary error function: erfc(x) = 1 - erf(x)
static Real	factorial(Real n)	Factorial function: n!
static Real	gamma(Real x)	Gamma function Γ(x) - generalization of factorial to real numbers.
static Real	hermite(int n, Real x)	Hermite polynomial H_n(x) evaluated at x (physicist's convention).
static Real	incompleteBeta(Real x, Real a, Real b)	Incomplete beta function: I_x(a, b).
static Real	incompleteGamma(Real a, Real x)	Incomplete gamma function: γ(a, x) = ∫₀ˣ t^(a-1) e^(-t) dt
static Real	laguerre(int n, Real x)	Laguerre polynomial L_n(x) evaluated at x.
static Real	legendre(int n, Real x)	Legendre polynomial P_n(x) evaluated at x.
static Real	logGamma(Real x)	Natural logarithm of Gamma function: ln(Γ(x))

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

SpecialFunctions

public SpecialFunctions()

Method Details

gamma

public static Real gamma(Real x)

Gamma function Γ(x) - generalization of factorial to real numbers.

Γ(n) = (n-1)! for positive integers Uses Lanczos approximation for accuracy.

logGamma

public static Real logGamma(Real x)

Natural logarithm of Gamma function: ln(Γ(x))

More numerically stable than Γ(x) for large x.

beta

public static Real beta(Real x,
 Real y)

Beta function: B(x, y) = Γ(x)Γ(y) / Γ(x+y)

Used in probability distributions (Beta, Dirichlet).

erf

public static Real erf(Real x)

Error function: erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt

Probability: P(X ≤ x) for normal distribution = ½[1 + erf(x/√2)] Uses Abramowitz invalid input: '&' Stegun approximation.

erfc

public static Real erfc(Real x)

Complementary error function: erfc(x) = 1 - erf(x)

besselJ0

public static Real besselJ0(Real x)

Bessel function of the first kind Jâ‚€(x).

Solutions to Bessel's differential equation. Used in physics: wave propagation, heat conduction, vibrations.

besselY0

public static Real besselY0(Real x)

Bessel function of the second kind Yâ‚€(x).

factorial

public static Real factorial(Real n)

Factorial function: n!

For non-integers, uses Γ(n+1).

legendre

public static Real legendre(int n,
 Real x)

Legendre polynomial P_n(x) evaluated at x.

Uses recurrence relation: P_0(x) = 1, P_1(x) = x, P_{n+1}(x) = ((2n+1)xP_n(x) - nP_{n-1}(x))/(n+1)

Parameters:

n - degree of the polynomial (n >= 0)

x - point at which to evaluate

Returns:

P_n(x)

hermite

public static Real hermite(int n,
 Real x)

Hermite polynomial H_n(x) evaluated at x (physicist's convention).

Uses recurrence relation: H_0(x) = 1, H_1(x) = 2x, H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)

Parameters:

n - degree of the polynomial (n >= 0)

x - point at which to evaluate

Returns:

H_n(x)

laguerre

public static Real laguerre(int n,
 Real x)

Laguerre polynomial L_n(x) evaluated at x.

Uses recurrence relation: L_0(x) = 1, L_1(x) = 1-x, L_{n+1}(x) = ((2n+1-x)L_n(x) - nL_{n-1}(x))/(n+1)

Parameters:

n - degree of the polynomial (n >= 0)

x - point at which to evaluate

Returns:

L_n(x)

incompleteGamma

public static Real incompleteGamma(Real a,
 Real x)

Incomplete gamma function: γ(a, x) = ∫₀ˣ t^(a-1) e^(-t) dt

Regularized: P(a, x) = γ(a, x) / Γ(a)

Parameters:

a - Shape parameter (a > 0)

x - Upper limit (x >= 0)

Returns:

Regularized incomplete gamma P(a, x)

incompleteBeta

public static Real incompleteBeta(Real x,
 Real a,
 Real b)

Incomplete beta function: I_x(a, b).

Regularized: I_x(a, b) = B(x; a,b) / B(a,b) Used in statistics: CDF of Beta distribution.

Parameters:

x - Value (0 invalid input: '<'= x invalid input: '<'= 1)

a - First parameter (a > 0)

b - Second parameter (b > 0)

Returns:

Regularized incomplete beta I_x(a, b)

airyAi

public static Real airyAi(Real x)

Airy function Ai(x).

Solution to y'' - xy = 0 that decays as x → ∞. Used in quantum mechanics (wave functions near turning points).

besselJn

public static Real besselJn(int n,
 Real x)

Bessel function of the first kind J_n(x) for integer order.

Parameters:

n - Order (integer)

x - Argument

Returns:

J_n(x)

besselYn

public static Real besselYn(int n,
 Real x)

Bessel function of the second kind Y_n(x) for integer order.

Parameters:

n - Order (integer, n >= 0)

x - Argument (x > 0)

Returns:

Y_n(x)

besselIn

public static Real besselIn(int n,
 Real x)

Modified Bessel function of the first kind I_n(x).

I_n(x) = i^(-n) * J_n(ix)

Parameters:

n - Order (integer)

x - Argument

Returns:

I_n(x)

besselKn

public static Real besselKn(int n,
 Real x)

Modified Bessel function of the second kind K_n(x).

Decays exponentially for large x, diverges at x=0.

Parameters:

n - Order (integer, n >= 0)

x - Argument (x > 0)

Returns:

K_n(x)