### 2.8 Exponentials in 3Di

#### 2.81 Exponential Graph Rotation

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.

#### 2.82 Lambert W Function

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