Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to [surreal numbers](http://mathworld.wolfram.com/SurrealNumber.html) or [surcomplex numbers](http://en.wikipedia.org/wiki/Surcomplex_number)? Which of their properties and relations (e.g. usual trig identities) will still hold in this case? Is it possible to perform differentiation and integration on these generalized functions?



Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?
The following is taken from a [sci.math](http://www.ics.uci.edu/~eppstein/cgt/surreal.html) post by Nicolau C Saldanha. Much more detail is available there. It is important to note that Nicolau does not define functions extensionally, i.e. two functions which agree everywhere but are defined differently are considered distinct.


We use the convention $f(x) = \{ f\_L(x,x\_L,x\_R) | f\_R(x,x\_L,x\_R) \}$ for surreal functions (note that this already breaks extensionality). We can define the definite integral of a surreal function as follows:


$$\begin{align}
\int\_a^b f(t) dt = \left\{\int\_a^{b\_L} f(t) dt + \oint\_{b\_L}^b {f\_L}(t) dt,\; \int\_a^{b\_R} f(t) dt + \oint\_{b\_R}^b {f\_R}(t) dt,\right.\\\\
 \int\_{a\_R}^b f(t) dt + \oint\_a^{a\_R} {f\_L}(t) dt,\; \int\_{a\_L}^b f(t) dt + \oint\_a^{a\_L} {f\_R}(t) dt\huge\mid\\\\ 
 \int\_a^{b\_L} f(t) dt + \oint\_{b\_L}^b {f\_R}(t) dt,\; \int\_a^{b\_R} f(t) dt + \oint\_{b\_R}^b {f\_L}(t) dt,\\\\ 
 \left.\int\_{a\_R}^b f(t) dt + \oint\_a^{a\_R} {f\_R}(t) dt,\; \int\_{a\_L}^b f(t) dt + \oint\_a^{a\_L} {f\_L}(t) dt\right\}
\end{align}$$


**Note:** $\oint$ denotes *direct integration*, meaning the integral is evaluated without chopping the domain into pieces, not countour integration. Properly the integral should have a capital $D$ superimposed on it, but I am not sure how to do this on this site.


The definition of integral gives you $\ln x$ right away: $\ln x = \int\_1^x \frac1t dt$. The integral also lets you solve the differential equations $y'=y$ and $y''=-y$, giving you $e^x,\sin x$ and $\cos x$. Nicolau claims that most standard functions, and even functions like the Riemann Zeta function should be obtainable in this manner.


However, this definition has its problems, which are discussed briefly in the linked sci.math thread. In particular, it does not behave nicely with the surreal exponential function. 


An exposition on the surreal exponential can be found in Harry Gonshor's "An introduction to the Theory of Surreal Numbers".


