A lot of math professionals rather likely still think that 3D complex numbers do not exist, may be for reasons like there are non-invertible numbers or whatever what other reason they have. This post more or less proved such views are nonsense; for example a lot of math on the 2D complex plane does not rely on the fact it is a field (and as such only division by zero is forbidden).

But on the 3D complex and circular number spaces indeed it brings some complications if you have non-invertible numbers in the function you want to integrate over a particular curve. And I have to say that problem could be solved by using the special properties that those numbers have. In this post I only show some examples with the non-invertible number alpha (alpha is the midpoint of the 3D exponential circles and all multiples of alpha are also non-invertible so the line through 0 and alpha are all not invertible).

For me writing this was a good distraction away from all that negative news we have day in day, all those countries reporting daily death toll can make you a bit depressed… So when I am through with the daily news I always do some other stuff like calculating a few of such integrals. That is a very good antidote against all that bad news. After all there is not much gained if you constantly think about things you cannot change at all.

This post is relatively long; at first I crafted 12 pictures but it soon turned out that was not enough. So while filling the 12 pictures with the math and the text I expanded some of the pictures so they could contain more math & text. That was not enough and in the end I had to craft two more background pictures. All in all it is 14 pictures long, that is a record length for this website.

If in your own mathematical life you have performed contour integration in the complex plane, you must be able to understand how this works in the 3D spaces. And for those who have done the thing known as *u*-substitution on the real line: it is just like that but now this *u* thing is the parametrization of a path. All that stuff below with gamma in it is either the path or the parametrization of that path. Please remark that you must use the complex or the circular multiplication on 3D, just like integrating over a contour in the complex plane uses the 2D complex multiplication.

In case if you are not familiar with the number alpha that is found at the center of the exponential circle, use the search function of this website and for example look up ‘seven properties of the number alpha’.

I hope I have removed all faults, typo’s etc so that later I do not have to repair the math because that is always cumbersome. Here we go: 14 pictures long so this is hard to grasp in detail in just a few hours. But it is beautiful math & that is why I do this. For me math is a lovely hobby.

Enough of the blah blah blah, here we go:

Ok, let´s first hit the button ´Publish´ and see what will happen…

It looks all right but a day after first publication I realized there was some missing text. Example two was not ‘cut & pasted’ in the above pictures.

So that was my fault, sorry if it made you confused.

Thus we need one more picture containing example 2:

Later I will flea through the rest of the text, if needed I will post more addenda. For the time being that was it so till addendums or till the next post.