# Define a function of multiple arguments at a given point

Trivia
I am using Taylor series expansions to solve a system of PDEs for a number of functions $$f_1(z, \phi), f_2(z, \phi) ...$$ The expansions are generated by Series command.

The functions are expanded in $$z$$ around $$z=0$$, and therefore, the coefficients of the series are functions wrt $$\phi$$, e.g. $$f_1^{(0,1)}[0,\phi]$$, $$f_1^{(1,2)}[0,\phi]$$. Here $$f_1^{(a,b)}[0,\phi]$$ reads as a-th order coefficient in the expansions of $$f_1[z,\phi]$$, differentiated $$b$$ times wrt $$\phi$$.

Question
Suppose that I have found $$f_1^{(a,0)}[0,\phi] = g[\phi]$$.
I want to be able to substitute it as a function, i.e. something like f1[0] -> Function[x, g[x] ], to evaluate expressions like
$$f_1^{(a,2)}[0,\phi]+(f_1^{(a,1)}[0,\phi])^2\qquad$$ to $$\qquad g''[\phi] + (g'[\phi])^2$$

How should I implement it?

## 1 Answer

Ok. Sorry, it was a trivial question

If we know that e.g. $$f_1^{(1,1)}[0,\phi] = g[\phi]$$ then

Derivative[1,n_Integer][f1][0, phi] :> D[g[phi], {phi,n}]