# Power series of dependent and independent variables

Let $f(z,w)$ be an analytic function in two variables where $w=w(z)$ is dependent on $z$ ($z$ is the independent variable). Then $f(z,w)$ has a power series expansion centered at $(z_0,w(z_0))$

$f(z,w)=\displaystyle\sum_{k,n=0}^\infty a_{k,n}(z-z_0)^k(w-w(z_0))^n$.

I've seen a general Taylor expansion for two independent variables using partial derivatives, i.e.

$f(x,y)=f(x_0,y_0)+[f_x(x_0,y_0)(x-x_0)+f_y(y-y_0)]+\frac{1}{2!}[f_{xx}(x-x_0)^2+2f_{xy}(x-x_0)(y-y_0)+f_{yy}(y-y_0)^2]+...$

Can I interpret this to give the power series where one variable is dependent of the other? What would the partial derivatives be?

In my opinion it does not make sense searching: How to transform an analytic function $f(x,y)$ of two variables to an analytic function $f(x,g(x))$ of one variable? E.g., take $f(x,y)= x+y$.
On the other hand, one can always ask whether an equation of type $f(x,y)=0$ implicitly defines a function $y=y(x)$ or $x=x(y)$. The theorem on implicit functions states: If the partial derivative $f_y(x_0,y_0) \neq 0$ then a neighbourhoud and a function $y=g(x)$ exist, such that $f(x,y)=0$ in the neighbourhoud iff $y=g(x)$. In this case $\frac {dg}{dx}(x)=-\frac{f_x(x,g(x))}{f_y(x,g(x))}$.