_WPRTLINK Complex atan( const Complex &z ) {
/*****************************************/
// Arctan of a complex number.
// From "Complex Variables and Applications" (pg. 63).
// atan( z ) = i/2 * log( (i + z)/(i - z) )
// = i/2 * log( (i + (z.r + z.i*i))/(i - (z.r + z.i*i)) )
// = i/2 * log( (z.r + (1+z.i)*i)/(-z.r + (1-z.i)*i) )
//
// let a = z.r, b = z.i
// (a + (1 + b)*i)/(-a + (1 - b)*i)
//
// or: a+(1+b)*i -a-(1-b)*i - a^2 + 1 - b^2 - (1 - b + 1 + b )*a*i
// ------------ * ------------ = ----------------------------------------
// -a+(1-b)*i -a-(1-b)*i a^2 + b^2 - 2*b + 1
//
// 1 - a^2 - b^2 - 2*a*i
// = -----------------------
// a^2 + b^2 - 2*b + 1
//
// Note: a^2 + b^2 == (a+b)^2 - 2ab
// 1 - (a+b)^2 + 2*a*b - 2*a*i
// = -----------------------------
// (a+b)^2 - 2*a*b - 2*b + 1
//
dcomplex value;
double apb2 = (z.real()+z.imag())*(z.real()+z.imag());
double _2ab = 2.0*z.real()*z.imag();
value = _IF_C16Div( 1.0 - apb2 + _2ab, -2.0*z.real(),
apb2 - _2ab - 2.0*z.imag() + 1.0, 0.0 );
value = _IF_CDLOG( value.realpart, value.imagpart );
return Complex( -value.imagpart / 2, value.realpart / 2 );
}
_WPRTLINK Complex asinh( const Complex &z ) {
/******************************************/
// Hyperbolic arcsin of a complex number.
// From "Complex Variables and Applications" (pg. 63).
// asinh( z ) = log( z + sqrt( z*z + 1 ) )
// = log( z + sqrt( 1 + (z.r + z.i)*(z.r - z.i) + 2*z.r*z.i*i ) )
dcomplex value;
value = _IF_CDSQRT( 1.0 + (z.real()+z.imag())*(z.real()-z.imag()),
2.0 * z.real() * z.imag() );
value = _IF_CDLOG( value.realpart + z.real(), value.imagpart + z.imag() );
return Complex( value.realpart, value.imagpart );
}
_WPRTLINK Complex log( const Complex &z ) {
/****************************************/
dcomplex result;
result = _IF_CDLOG( z.real(), z.imag() );
return Complex( result.realpart, result.imagpart );
}
