public delegate double fPtr_Jordan_parametriz_(double t); // the x(t) or y(t) or dx(t) or dy(t) of the Jordan parametric contour.
#region exampleFunctions
//private static double x( double t)
//{// x(t)=t
// return t;
//}// x(t)
//private static double y( double t )
//{// y(t)=2*t+1
// return 2 * t + 1;
//}// y(t)
//private static double dx( double t )
//{// x(t)=t->dx(t)=x'(t)dt=1*dt
// return +1.0;
//}// x(t)
//private static double dy( double t )
//{// y(t)=2*t+1 ->dy(y)=y'(t)dt==2*dt
// return +2.0;
//}// y(t)
///// <summary>
///// the functions choosen for the example are f(z)=z which implies u(x,y)=x, v(x,y)=y;
///// the choice for the contour is x(t)=t,y(t)=2*t+1,dx=dt which means dx=1
///// dy=2*dt which means dy=2.
///// </summary>
//private static double genericIntegrand_u_part( double x, double y )
//{// f(z)==z -> Re(f(z))==Re(f(x+I*y))==x.
// return x;
//}// u(x,y)
////
//private static double genericIntegrand_v_part( double x, double y )
//{// f(z)==z -> Im(f(z))==Im(f(x+I*y))==y.
// return y;
//}// v(x,y)
#endregion exampleFunctions
/// <summary>
/// Trapezium Integration. NB.: (u(x,y)+I*v(x,y) )*(dx+I*dy)==u*dx-v*dy + I*( u*dy+v*dx)
/// ...
/// </summary>
private static double Integrate_equi_trapezium_RealPart(
double t0, double tn, // extrema in the pull-back t in [t0,tn]
fPtr_U_or_V_ u_Part,
fPtr_U_or_V_ v_Part,
fPtr_Jordan_parametriz_ x_coordinate,
fPtr_Jordan_parametriz_ y_coordinate,
fPtr_Jordan_parametriz_ dx_differential,
fPtr_Jordan_parametriz_ dy_differential,
Int64 n) // #trapezia in the partition
{ // RealPart == ==u*dx-v*dy
double DeltaT = (tn - t0) / (double)n,
res = 0.0,
t = DeltaT;// the boundaries {t0,tn} are computed separately, after the core-loop. So trapezium starts at 1*DeltaT.
// kordell starts here.
for (; t < tn; t += DeltaT) // stop at the second to last, i.e. <tn. The boudaries are computed separately: t=t0, t=tn.
{ // sum all the internal sides
res += u_Part(x_coordinate(t), y_coordinate(t)) * dx_differential(t) - v_Part(x_coordinate(t), y_coordinate(t)) * dy_differential(t);
}
// post kordell adjustments
res *= DeltaT; // multiply them for the common base
res += (
u_Part(x_coordinate(t0), y_coordinate(t0)) * dx_differential(t0) - v_Part(x_coordinate(t0), y_coordinate(t0)) * dy_differential(t0) +
+u_Part(x_coordinate(tn), y_coordinate(tn)) * dx_differential(tn) - v_Part(x_coordinate(tn), y_coordinate(tn)) * dy_differential(tn)
) * 0.5 * DeltaT; // add extrema * base/2
// ready
return(res);
}//
}//
private static bool extremaCheck(
ComplexField.Complex z0,
ComplexField.Complex z1,
double t0, double tn, // extrema in the pull-back
fPtr_Jordan_parametriz_ x_coordinate,
fPtr_Jordan_parametriz_ y_coordinate
)
{
double localEpsilon = +1.0E-13;
double z0_x_byJordan = x_coordinate(t0);
double z0_y_byJordan = y_coordinate(t0);
double z1_x_byJordan = x_coordinate(tn);
double z1_y_byJordan = y_coordinate(tn);
//
double delta_Z0x = Math.Abs(z0_x_byJordan - z0.get_real);
double delta_Z0y = Math.Abs(z0_y_byJordan - z0.get_immaginary);
double delta_Z1x = Math.Abs(z1_x_byJordan - z1.get_real);
double delta_Z1y = Math.Abs(z1_y_byJordan - z1.get_immaginary);
//
bool res = (
delta_Z0x < localEpsilon &&
delta_Z0y < localEpsilon &&
delta_Z1x < localEpsilon &&
delta_Z1y < localEpsilon
);
//ready.
return(res);
}// extremaCheck
}// extremaCheck
/// <summary>
///
/// </summary>
/// <param name="sigma">the Real part of the complex argument</param>
/// <param name="t">the Immaginary part of the complex argument</param>
/// <param name="xRplus_threshold">the x0 in R+ used as stop-value in the improper integration to +Infinity</param>
/// <param name="n">the number of DeltaX to step into, i.e. the number of trapezia to be calculated in the decomposition</param>
/// <returns>the Complex value of the Gamma[sigma+I*t]</returns>
public static ComplexField.Complex ContourIntegral_ManagementMethod(
ComplexField.Complex z0,
ComplexField.Complex z1,
double t0, double tn, // extrema in the pull-back
fPtr_U_or_V_ u_Part,
fPtr_U_or_V_ v_Part,
fPtr_Jordan_parametriz_ x_coordinate,
fPtr_Jordan_parametriz_ y_coordinate,
fPtr_Jordan_parametriz_ dx_differential,
fPtr_Jordan_parametriz_ dy_differential,
Int64 n) // #trapezia in the partition
{
bool extremaAdequacy = extremaCheck(
z0, z1,
t0, tn,
x_coordinate,
y_coordinate
);
if (!extremaAdequacy)
{
throw new System.Exception("Integration extrema do not match, between coChain and Jordan-path.");
}
ComplexField.Complex res = new ComplexField.Complex(
Integrate_equi_trapezium_RealPart(
t0, tn
, u_Part
, v_Part
, x_coordinate
, y_coordinate
, dx_differential
, dy_differential
, n)
, Integrate_equi_trapezium_ImmaginaryPart(
t0, tn
, u_Part
, v_Part
, x_coordinate
, y_coordinate
, dx_differential
, dy_differential
, n)
);
// ready.
return(res);
} // ContourIntegral_ManagementMethod
