I am currently reading a German book about complex analysis $($to be precise K. Fritzsche: Grundkurs Funktionentheorie$)$. Within the $4$th chapter he deals with elliptic functions and especially with the Weierstrass $\wp$ function. There he writes
Let $\mathbb C(z)$ be the field of rational functions on $\mathbb C$ then we can define a ring homomorphism $\varphi:\mathbb C(z)[x]\to K(\Gamma)$ as $$\varphi\left(\sum_{i=0}^N R_i(x)x^i\right):=\sum_{i=0}^N R_i(\wp)\cdot(\wp')^i.$$
I am not sure whether I really understand the notation which is used here. I am used to $\mathbb F[x]$ refering to the polynomials with coefficients in $\mathbb F$. Does the notation $\mathbb C(z)[x]$ simply refers to all polynomials with coefficients rational functions over $\Bbb C$ in $z$? This would match the later defined ring homomorphism $\varphi$ as $R(x)$ commonly denotes the rational functions in $x$ over $R$.
Is my reasoning correct, i.e. does $\mathbb C(z)[x]$ denotes all polynomials of $x$ with coefficients in $\mathbb C(z)$, the field of rational functions in $z$ over $\Bbb C$, or am I missing something? Moreover is this a standard notation hence I have never seen a construct like this before.
Thanks in advance!
You are correct: $\mathbb C(z)[x]$ denotes all polynomials in $x$ with coefficients in $\mathbb C(z)$, the field of rational functions in $z$.