Taylor-like expansion for multivariable functions Is there any analogue for taylor series for multivariable functions? In other words, can we rewrite any function as a sum of algebraic terms? For example, $x^y$. Can it be written of the form $\sum C_{m,n}x^my^n$, where $C_{m,n}$ is some constant pertaining to the particular m,n (most probably in terms of $\frac{\partial^m}{\partial x^m}x^y$ etc). Is there a generalization for more then two variables? Another example would be $\frac{x+y}{x^2+y^2}$. I suspect that it can be derived by using partial differentials and mashing together the taylor expansions of $f(x,constant)$ and $f(constant,y)$, but I can't manage to do it.

Yes, such a thing exists and is well-known. Any good book on functions of several variables will have it. This article has a sketch of it.