45.3 Special Functions

Special functions arise in probability, statistics, and physics. Java lacks built-ins; implement or use libraries.

45.3.1 Gamma Function

Γ(n) = (n-1)! for integers; extends factorial to reals.

// Stirling's approximation (rough)
double gamma(double x) {
  return Math.sqrt(2 * Math.PI / x) * Math.pow(x / Math.E, x);
}

// For production, use Apache Commons Math or similar

45.3.2 Error Function (erf)

Used in probability (normal distribution CDF).

// Abramowitz & Stegun approximation
double erf(double x) {
  double a1 =  0.254829592;
  double a2 = -0.284496736;
  double a3 =  1.421413741;
  double a4 = -1.453152027;
  double a5 =  1.061405429;
  double p  =  0.3275911;

  int sign = x < 0 ? -1 : 1;
  x = Math.abs(x);
  double t = 1.0 / (1.0 + p * x);
  double y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t * Math.exp(-x*x);
  return sign * y;
}

Complement: erfc(x) = 1 - erf(x)

45.3.3 Bessel Functions

Solutions to Bessel's differential equation; appear in wave propagation.

// J₀(x) series expansion (small x)
double besselJ0(double x) {
  double sum = 1.0;
  double term = 1.0;
  for (int k = 1; k < 20; k++) {
    term *= -(x*x) / (4.0 * k * k);
    sum += term;
    if (Math.abs(term) < 1e-10) break;
  }
  return sum;
}

For robust implementations, use specialized libraries (Apache Commons Math, JAMA).

45.3.4 Library Recommendations

• Apache Commons Math: comprehensive special functions
• JScience: physics/engineering functions
• Colt: high-performance scientific computing