{Real Special functions
authors: Nikolai V. Shokhirev and Eugene B. Krissinel )
created: 1991.09.09
modified: 2002.01.01
last modified: 2003.02.02
ŠNikolai V. Shokhirev, 2002-2003 }
unit uSpecFunc;

interface

uses
  uMatTypes;

const
  ln10     = 2.302585092994046; // 2.3025850929940456840179915
  Gamma_Eu = 0.577215664901532860606512;

{ obsolete }
function ArcSin(x: TFloat): TFloat;
function ArcCos(X: TFloat): TFloat;
{ Hyperbolic Functions }
function ch(x: TFloat)    : TFloat;
function sh(x: TFloat)    : TFloat;
function th    (x: TFloat): TFloat;
function arch  (x: TFloat): TFloat;
function arsh  (x: TFloat): TFloat;
function arth  (x: TFloat): TFloat;

{ Special functions for the Real Calculations }
{ er(x) = exp(x**2)*erfc(x)    }
function er(x: TFloat)        : TFloat;
{ error function: 1 = erfc+erf }
function erf(x: TFloat)       : TFloat;
{ complementary error function }
function erfc(x: TFloat)      : TFloat;
{ Modified Bessel function     }
function I0_(x: TFloat)       : TFloat;
{ Modified Bessel function     }
function I1_(x: TFloat)       : TFloat;
{ Modified Bessel function     }
function K0_(x: TFloat)       : TFloat;
{ Modified Bessel function     }
function K1_(x: TFloat)       : TFloat;
{ fk0(x)=K0(x)+ln(x/2)*I0(x)   }
function fk0_(x: TFloat)      : TFloat;
{ fk1(x)=K1(x)-ln(x/2)*I1(x)   }
function fk1_(x: TFloat)      : TFloat;

{ Gamma-function  error < 5 e-7  }
function Gamma_(x: TFloat)    : TFloat;
{ Gamma-function   error < 5 e-16 , x <> 0,-1,-2, .. }
function Gamma(X: TFloat): TFloat;
{ incomplete Gamma-function   error < 5 e-16
 GIM = Gamma(A,X)  int(X < t < inf )
 GIN = gamma(A,X)  int(0 < t < X )
 GIP = P(A,X), gamma(A,X) = P(A,X)*Gamma(A)
 Gamma(A) = GIN + GIM }
procedure INCOG(A, X: TFloat; var GIN, GIM, GIP: TFloat);
{ incomplete gamma-function   error < 5 e-10  }
function gammaInc(a,x: TFloat): TFloat;
{ incomplete gamma-function   error < 5 e-10 gammaX(a,-x) = -gammaX(a,x) }
function  gammaX(a,x: TFloat): TFloat;
{ PolyGamma(n,z) = d[n+1]ln(Gamma(z))/dz[n+1] ,  z > 10, 0 <= n <= 3  }
function  PolyGamma(n: TInt; z: TFloat): TFloat;(*)

{ integral exp-function        }
function E1_(x: TFloat)       : TFloat;
{ Modified Bessel function     }
function I_nu(nu, x: TFloat)  : TFloat;

(*)
{ Undocumented Functions }
//function Gamma(x: TFloat): TFloat;
(*)function  BJn   (n,x: TFloat): TFloat;
function  BIn   (n,x: TFloat): TFloat;
function  BKn   (n,x: TFloat): TFloat;
function  Sn(n: TInt; z: TFloat): TFloat;
function  Un(n: TInt; z,a: TFloat): TFloat;
(*)
{ ===================================================================  }

implementation

end.

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