I see a number of formulations, but I don't see a rigorous treatment of why those formulations deserve those particular names. Which properties from the 1D space, and which properties from other multi-dimensional variants of those functions, are preserved?

For one illustrative example, is the derivative of your sine function, the cosine function, for all D? If so, which derivative? What about the partial derivatives?

I'm not saying you're wrong, I'm just saying it's not obvious why those functions are correct analogs, and I'm not going to do all the work to derive all that myself when you didn't in the original paper :-)