Hallo
allgemeine Lösungsform für z^2 = re + im i
z1 = r^0.5 * e^{i*phi/2} sowie
z2 = r^0.5 * e^{i*(phi/2 +180°)}
z1^2 = r*e^iphi = r * (cos phi) + i * r * (sinphi) = 1 - i
r1*cos phi1 = 1
r1*sin phi1 = -1
Teilen ergibt : (r1*sin phi1) / (r1*sin phi1) = tan phi1 = -1/1 = -1 => phi1 = -45°
Daraus folgt r1 = -1/(sin phi1) = -1/(1/2)^0.5 = 2^0.5
=> z1 = 2^0.25 * e^{i* (-22.5°)} = 2^0.25 * (cos -22.5 +i*sin -22.5) = 2^0.25 *(cos +22.5° -i*sin +22.5°)
= 2^0.25 *( (1/2 +1/2*cos 45°)^0.5) - i * 2^0.25*( (1/2 -1/2 * cos 45°)^0.5 )
= 2^0.25 *( (1/2 +1/2 * (1/2)^0.5)^0.5) - i * 2^0.25*( (1/2 -1/2 * (1/2)^0.5)^0.5 )
= (1/2 * 2^0.5 +1/2 )^0.5 - i * (1/2 * 2^0.5 -1/2)^0.5
z1 = 1.09864113 - i * 0.45508986=> z1^2 = 1.09864113^2 -0.45508986^2 -i * 2*0.45508986*1.09864113 = 1 -i
=> z2 = 2^0.25 * e^{i* (-22.5° +180°)} = 2^0.25 * e^{i*(157.5°)} = 2^0.25 * (cos 157.5 +i*sin 157.5)
= 2^0.25 *( - (1/2 +1/2*cos 45°)^0.5) + i * 2^0.25*( (1/2 -1/2 * cos 45°)^0.5 )
= 2^0.25 *(- (1/2 +1/2 * (1/2)^0.5)^0.5) + i * 2^0.25*( (1/2 -1/2 * (1/2)^0.5)^0.5 )
= - (1/2 * 2^0.5 +1/2 )^0.5 + i * (1/2 * 2^0.5 -1/2)^0.5
z2 = - 1.09864113 + i * 0.45508986
=> z2^2 = (-1.09864113)^2 -0.45508986^2 -i * 2*0.45508986*1.09864113 = 1 -i