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The Complex Exponential 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 f(z)=e^z, where e is the number 2.718281828459.... In coordinates, if z=x+iy, then e^z=e^xe^(iy)=e^x(cos(y)+Isin(y)).