Acosc1.cin

// Acosc1.cin is set of routines for evaluation acosc1, whih is first branch of function ArcCosc behind the cut line at the negative part of the real axis.

z_type acos(z_type z){ if(Im(z)<0){if(Re(z)>=0){return I*log( z + sqrt(z*z-1.) );} else{return I*log( z - sqrt(z*z-1.) );}} if(Re(z)>=0){return -I*log( z + sqrt(z*z-1.) );} else {return -I*log( z - sqrt(z*z-1.) );} }

z_type cosc(z_type z) {return cos(z)/z;} z_type cosp(z_type z) {return (-sin(z) - cos(z)/z)/z ;} z_type cohc(z_type z) {return cosh(z)/z ;} z_type cohp(z_type z) {return (sinh(z)-cosh(z)/z)/z ;}

z_type acoscL(z_type z){ int n; z_type s,q; z*=-I; q=I*sqrt(1.50887956153832-z); s=q*1.1512978931181814 + 1.199678640257734; DO(n,6) s+= (z-cohc(s))/cohp(s); return -I*s; }

z_type acoscR(z_type z) {int n; z_type s= (1.-0.5/(z*z))/z; DO(n,5) s+=(z-cosc(s))/cosp(s); return s;}

z_type acoscB(z_type z){ z_type t=0.33650841691839534+z, u=sqrt(t), s; int n; s=   2.798386045783887 +u*(-2.437906425896532 +u*( 0.7079542331649882 +u*(-0.5009330133042798 +u*( 0.5714459932734446   )))); DO(n,6) s+=(z-cosc(s))/cosp(s); return s; }

DB Sazae0= 2.798386045783887; DB Tarao0=-0.33650841691839534; DB Sazae1= 6.1212504668980685; DB Tarao1= 0.161228034325064;

z_type acosc(z_type z){DB x1=-0.33650841691839534,x=Re(z),y=Im(z),yy=y*y,r; r=x-x1;r*=r;r+=yy; if(r < 1.8 )    return acoscB(z); r=x+2.;r*=r;r+=yy; if(r>8. && x>=0) return acoscR(z); if(y >= 0) return acoscL(z); return conj(acoscL(conj(z))); }

z_type acosc1R(z_type z){int n; z_type s,t,u; t=Tarao1-z; u=sqrt(t); s=Sazae1- 3.522043522040332*u; s+=(z-cosc(s))/cosp(s); DO(n,5) s+=(z-cosc(s))/cosp(s); return s; }

z_type acosc1B(z_type z){ z_type t=0.33650841691839534+z, u=sqrt(t), s; int n; s=   2.798386045783887 +u*( 2.437906425896532 +u*( 0.7079542331649882 +.16*u*( 0.5009330133042798 // I have no idea about this.. +u*( 0.5714459932734446   )))); DO(n,6) s+=(z-cosc(s))/cosp(s); return s; }

z_type acosc1H(z_type z){int n; z_type s=2*M_PI - acos(2*M_PI*z); DO(n,6) s+=(z-cosc(s))/cosp(s); return s; }

z_type acosc1(z_type z){ DB x=Re(z), y=Im(z); if( abs(z) > 1 ) return acosc1H(z); if( x < 0 )      return acosc1B(z); return acosc1R(z); }

// The acosc1(z) returns of order of 14 correct decimal digits while \(|z|<6 \). // Let me know if any problem with the use. // Copileft 2012 by Dmitrii Kouznetsov