Function theory of a hyperbolic variable

2011-07-30T01:04:21Z

I've found quite a number of articles on the basics of function theory in one hyperbolic (split-complex, dual, duplex, motro,..) variable, perhaps the most notable being http://arxiv.org/PS_cache/math-ph/pdf/0507/0507053v2.pdf. What is covered in this and the other articles are more or less only the counterparts of most elementary topics in complex analysis. Are there really no deeper results in this theory or are they just so hard to find? If so, some links or summaries of results would be dearly appriciated.

Particularly http://clifford-algebras.org/v8/81/MOTTER81.pdf aksed for a hyperbolic equvalence of Cauchy integral formula. Thus two rather different answers were provided in http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.0375v1.pdf and http://www.springerlink.com/content/kp44rl074g7187n2 . Both articles mention immense applications that follow directly from their formulas, but I havn't been unable to find a single article discussing them.

And it is quite apparent that hyperbolic Cauchy-like formulas don't yield the percise same results as in complex analysis since it is easy to show not only that hyperbolic holomorphic functions are not allways analytic, but that they need not be even $C^2$! So could somebody explain what those mentioned direct implications of hyperbolic Cauchy formulas and also explain why are there two formulas to begin with? Thank you!

Answer by paul garrett for Function theory of a hyperbolic variable

2011-07-30T13:31:37Z

The problem is that "adjoining" a "number" $j$ to $\mathbb R$ such that $j^2=+1$ rather than $i^2=-1$ gives, by Sun-Ze's theorem, $$ \mathbb R[j] \approx \mathbb R[x]/\langle x^2-1\rangle \approx \mathbb R[x]/\langle x-1\rangle \oplus \mathbb R[x]/\langle x+1\rangle \approx \mathbb R\oplus \mathbb R $$ In particular, there are $0$-divisors, such as $(0,1)\cdot (1,0)=(0,0)$. A corresponding change of coordinates gives a analogue of the Cauchy-Riemann operator: just $\frac{\partial}{\partial x}\pm \frac{\partial}{\partial y}$.

Notably, these two operators are the factors of the one (spatial) dimensional wave equation, whose analytic features/failings caused so much consternation/interest pre-1800, namely, any function of the form $F(x,y)=f(x-y)$ is apparently annihilated by $\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$, even when $f$ is not as smooth as one might think it ought to be.

This is in extreme contrast to the $i^2=-1$ situation, where the Cauchy-Riemann operator's vanishing gives a definite constraint (a.k.a. "ellipticity").

Edit: [thx for helpful edit-corrections...] If one uses the "usual" norm/length in the denominator, in the definition of "derivative", this does avoid zero-divisors, but changes the thing to being something else entirely, I think.

The fact that $\mathbb R$ with $j$ such that $j^2=+1$ adjoined is not a field is inescapable, and quite unlike the case of complex numbers. On another hand, it is certainly true that Clifford analysis is useful, if not quite in this fashion. Yes, "Dirac operators" have many roles, as factoring second-order differential operators into first-order. Yes, the Laplacian in $\mathbb R^2$ factors into Cauchy-Riemann and its conjugate, and the one-dimensional wave equation (as noted above) factors into two "real" linear operators. This is the goal, actually. For higher dimensions, these operators cannot factor into scalar differential operators, but Clifford-algebra-valued ones. Nevertheless, I don't think it's quite that the underlying "scalars" are made more exotic, but, rather, are enhanced by allowing operator-valued functions.

Function theory of a hyperbolic variable - MathOverflow

Function theory of a hyperbolic variable

Answer by paul garrett for Function theory of a hyperbolic variable