The Complex Square Root Function

Root, Square
Drag the blue points to see the effect of inverting the complex number a. The checkboxes show different shapes. The "before" shape is filled in, and is traced by the blue point P. The "after" shape is not filled, and is traced by P'.
This function finds the square root of a complex number z=x+iy. It can be computed by multiplying out (x+iy)^2. As with real numbers, square root is a 2-valued function: each complex has two square roots, with opposite signs. The choice of one of these signs defines what is called a branch of the square root function, which is a single-valued function. The absolute value of z^2 is the positive square root of the absolute value of z. The angle of z^2 is half the angle of z for the principal branch (Branch = +1). For the other branch, add 180 degrees to rotate to the other half of the plane.