A nomenclature issue in Complex Analysis, what is difference between between writing "f(z)" and "f"? As stated in the subject, this is just a curiosity about the nomenclature that is used in some Complex Analysis textbooks (particularly in problem statements).  I would think that writing "f(z)" would be the most general method of indicating f is a function in the complex plane.  Does simply writing "f" mean the same thing as "f(z)" or am I missing out on something serious here?
Thanks!

Technically, the function is $f$, not $f(z)$; here, $f(z)$ is the value of the function $f$ at some complex number $z$. So $f$ is the function (which is not a complex number), and $f(z)$ is a value of that function (which is a complex number).
However, writing $f(z)$ rather than $f$ usually (but not always!) causes no confusion. In my own opinion, you should write $f$ to be correct. 
Question: Why do many mathematicians write "$f(z)$" rather than simply "$f$" if the latter is more correct? 
Here's my opinion. Every time one introduces a function $f$, he/she must mention three items: the domain of $f$; the codomain of $f$; the value of $f$ for each $z$ in its domain. Now, mathematicians usually avoid giving all this info by simply stating $f(z)$ for every $z$ in the domain of $f$, and they assume the reader understands that the domain of $f$ is taken to be "the set of all $z$ for which $f(z)$ makes sense." (They also assume the reader will understand that the codomain is some large set. For complex-valued $f$, it's usually all of $\mathbb{C}$.) So after giving $f(z)$, all three items are known. No harm done.
