The Complex Logarithm Function

Drag the blue points to see the effect of adding the complex number a to various shapes. 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 applet shows the function , the natural logarithm of z. It is the inverse of the complex exponential function. It can be computed from the polar coordinates of z: . The logarithm is a multi-valued function, because the angle can be represented in many ways: is the same angle as , but it gives a different imaginary part for . To make single valued, choose a branch: a range for angles.