### A) New Helix Antiderivatives

Some typical antiderivatives like are extended to include complex numbers in both first and second level exponents and result in helixes.

One of the standard integral forms for exponentials is:

This can be generalized to complex first level exponents, complex second level exponents with real numbers in addition to the integers. For example:

Animation 46 ’Helix Antiderivative’

### B) A New Surface Geometry of Complex Natural Logarithms

With the new concept of rotating the usual exponential function, complex logs are now seen to exist as a four dimensional function. A complex number input to the function and a complex number output. The fourth dimension being the imaginary rotation.

The usual e-base exponential graph:

When given a complex exponent:

Will *rotate *the exponential graph in accordance with the value of

Animation 34 ‘e-base Exponential Rotation’

The rotating graph may be made into a surface by switching from ‘line-angle’ input to ‘surface input. I.e.

With differing ‘surface operators’ the geometry of the exponential surface may be altered, in this case to a ‘square.’

In addition to the path of a point, which is the grey ball below on the exponential graph that will trace out a ‘square,’ the actual rotation of the graph undergoes changes in its ‘rotational velocity.’ It decelerates at the corners of the ‘square’ and accelerates along the sides of the ‘square.’

Animation 69 ‘Square Rotating Exponential’

In both the line-angle case and the surface case, the complex natural log function will describe a point with coordinates or related by: