The usual exponential function that shows up in two dimensions has equation:
In 3Di this equation would be:
with for all values of If we were to add an arbitrary imaginary constant, say to the exponent:
the two dimensional graph, i.e. the FRP projection in red, would then be:
The two dimensional RSIP (right side imaginary plane) projection for this red graph is:
And in three dimensions:
Normal exponential graph in black.
Normal exponential graph with imaginary constant, in red.
The effect of adding the imaginary constant to the exponent is to rotate the graph in the positive imaginary, or depth dimension.
In three dimensions think of an ‘i’rotation as going from the horizontal or vertical to the front, rather than from the horizontal to the vertical.
This forms the basis of a new geometric interpretation of complex logarithms. See section 6.52 Rotating Exponentials.
If we were to add a complex coefficient to the equation, e.g.
the three dimensional graph, in blue, is altered to:
This blue graph in TIP (top Imaginary plane) projection with coordinates has the graph:
If the arbitrary imaginary constant in the exponent is changed to:
and the axes of the graph are reversed, i.e. , the Lambert W function graph results:
Inserting values for into equation:
shows that, using only the imaginary values, in two dimensions with the TIP axes reversed, the graph passes through the points .
The Omega constant itself, on this graph, is at the point