/// <summary>
/// This function returns the complex hyperbolic arccosine of
/// the real number a, arccosh(a).
/// </summary>
/// <param name="a">Function argument.</param>
/// <returns>The complex hyperbolic arccosine of
/// the real number a.</returns>
public static Complex Acosh(double a)
{
Complex z;
if (a >= 1)
{
z = new Complex(RMath.Acosh(a), 0);
}
else
{
if (a >= -1.0)
{
z = new Complex(0, Math.Acos(a));
}
else
{
z = new Complex(RMath.Acosh(-a), M_PI);
}
}
return(z);
}
/// <summary>
/// This function returns the complex arcsecant of the real number a,
/// arcsec(a) = arccos(1/a).
/// </summary>
/// <param name="a">Function argument.</param>
/// <returns>The complex arcsecant of the real number a.</returns>
public static Complex Asec(double a)
{
Complex z;
if (a <= -1.0 || a >= 1.0)
{
z = new Complex(Math.Acos(1 / a), 0.0);
}
else
{
if (a >= 0.0)
{
z = new Complex(0, RMath.Acosh(1 / a));
}
else
{
z = new Complex(M_PI, -RMath.Acosh(-1 / a));
}
}
return(z);
}
/// <summary>
/// This function returns the complex arccosine of the real number a, arccos(a).
/// </summary>
/// <param name="a">The function argument.</param>
/// <returns>The complex arccosine of the real number a.</returns>
/// <remarks>
/// For a between -1 and 1, the
/// function returns a real value in the range [0,pi]. For a
/// less than -1 the result has a real part of pi/2 and a
/// negative imaginary part. For a greater than 1 the result
/// is purely imaginary and positive.
/// </remarks>
public static Complex Acos(double a)
{
Complex z;
if (Math.Abs(a) <= 1.0)
{
z = new Complex(Math.Acos(a), 0);
}
else
{
if (a < 0.0)
{
z = new Complex(Math.PI, -RMath.Acosh(-a));
}
else
{
z = new Complex(0, RMath.Acosh(a));
}
}
return(z);
}
//-----------------------------------------------------------------------------------
//-----------------------------------------------------------------------------------
/// <summary>
/// The modulus (length) of the complex number
/// </summary>
/// <returns></returns>
public double GetModulus()
{
return(RMath.Hypot(Re, Im));
}
/// <summary>
/// This function returns the complex arcsine of the complex number a,
/// arcsin(a)}. The branch cuts are on the real axis, less than -1
/// and greater than 1.
/// </summary>
/// <param name="a">The function argument.</param>
/// <returns>the complex arcsine of the complex number a.</returns>
public static Complex Asin(Complex a)
{
double R = a.Re, I = a.Im;
Complex z;
if (I == 0)
{
z = Asin(R);
}
else
{
double x = Math.Abs(R), y = Math.Abs(I);
double r = hypot(x + 1, y), s = hypot(x - 1, y);
double A = 0.5 * (r + s);
double B = x / A;
double y2 = y * y;
double real, imag;
const double A_crossover = 1.5, B_crossover = 0.6417;
if (B <= B_crossover)
{
real = Math.Asin(B);
}
else
{
if (x <= 1)
{
double D = 0.5 * (A + x) * (y2 / (r + x + 1) + (s + (1 - x)));
real = Math.Atan(x / Math.Sqrt(D));
}
else
{
double Apx = A + x;
double D = 0.5 * (Apx / (r + x + 1) + Apx / (s + (x - 1)));
real = Math.Atan(x / (y * Math.Sqrt(D)));
}
}
if (A <= A_crossover)
{
double Am1;
if (x < 1)
{
Am1 = 0.5 * (y2 / (r + (x + 1)) + y2 / (s + (1 - x)));
}
else
{
Am1 = 0.5 * (y2 / (r + (x + 1)) + (s + (x - 1)));
}
imag = RMath.Log1p(Am1 + Math.Sqrt(Am1 * (A + 1)));
}
else
{
imag = Math.Log(A + Math.Sqrt(A * A - 1));
}
z = new Complex((R >= 0) ? real : -real, (I >= 0) ? imag : -imag);
}
return(z);
}