/////////////////////////////////////////
public static BigNum Gamma(BigNum z)
{
// http://www.rskey.org/gamma.htm
// n! = nn√2πn exp(1/[12n + 2/(5n + 53/42n)] – n)(1 + O(n–8))
// which requires the exponential function, and some function O (which the webpage fails to define, hmmm)
// so here's the infinite product series from Wikipedia instead
// Gamma(z) == (1/z) * Prod(n=1;n<inf;n++) { (1+1/n)^z / (1+z/n) }
const Int32 iterations = 25;
BigNumFactory f = z.Factory;
BigNum ret = z.Power(-1);
for (int n = 1; n < iterations; n++)
{
Double numerator1 = 1 + (1 / n);
BigNum numerator2 = Pow(f.Create(numerator1), z);
BigNum denominato = f.Unity + z / f.Create(n);
BigNum prodThis = numerator2 / denominato;
ret *= prodThis;
}
return(ret);
}
/// <summary>Computes the Natural Logarithm (Log to base E, or Ln(x)) of the specified argument</summary>
public static BigNum Log(BigNum x)
{
BigNumFactory f = x.Factory;
if (x <= f.Zero)
{
throw new ArgumentOutOfRangeException("x", "Must be a positive real number");
}
const int iterations = 25;
BigNum one = f.Unity;
BigNum two = f.Create(2);
// ln(z) == 2 * Sum(n=0;n<inf;n++) { (1 / (2n+1) ) * Pow( (z-1)/(z+1), 2n+1) }
BigNum ret = f.Zero;
for (int n = 0; n < iterations; n++)
{
int denom = 2 * n + 1;
Double coefficient = 1 / denom;
BigNum otherHalf = Pow((x - one) / (x + one), denom); // hurrah for integer powers
BigNum sumThis = f.Create(coefficient) * otherHalf;
ret += sumThis;
}
return(ret);
}
public static BigNum Sin(BigNum theta)
{
BigNumFactory f = theta.Factory;
// calculate sine using the taylor series, the infinite sum of x^r/r! but to n iterations
BigNum retVal = f.Zero;
// first, reduce this to between 0 and 2Pi
if (theta > f.TwoPi || theta < f.Zero)
{
theta = theta % f.TwoPi;
}
Boolean subtract = false;
// using bignums for sine computation is too heavy. It's faster (and just as accurate) to use Doubles
#if DoubleTrig
Double thetaDbl = Double.Parse(theta.ToString(), Cult.InvariantCulture);
for (Int32 r = 0; r < 20; r++) // 20 iterations is enough, any more just yields inaccurate less-significant digits
{
Double xPowerR = Math.Pow(thetaDbl, 2 * r + 1);
Double factori = BigMath.Factorial((double)(2 * r + 1));
Double element = xPowerR / factori;
Double addThis = subtract ? -element : element;
BigNum addThisBig = f.Create(addThis);
retVal += addThisBig;
subtract = !subtract;
}
#else
for (Int32 r = 0; r < _iterations; r++)
{
BigNum xPowerR = theta.Power(2 * r + 1);
BigNum factori = Factorial(2 * r + 1);
BigNum element = xPowerR / factori;
retVal += subtract ? -element : element;
subtract = !subtract;
}
#endif
// TODO: This calculation generates useless and inaccurate trailing digits that must be truncated
// so truncate them, when I figure out how many digits can be removed
retVal.Truncate(10);
return(retVal);
}