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!
You have two conflicting traditions of writing functions here:
The point is really that complex analysis is such an old-fashioned subject (there are perfectly useable textbooks on it written in the 1890s and 1920s) that to a certain extent it has remained traditionally immune to the new formalism of notation. Hence in complex analysis $f(z)$ is a function $\mathbb{C} \supseteq D \to \mathbb{C}$ of a complex variable $z$. This is also rather similar to the abstract algebra tradition of using $X$ as an "indeterminant": not a specific value in the original algebraic object, but an extra symbol that we may give a value later if we so choose, but not immediately.
(Personal anecdote: I know a logician who, due to his unusual route into Mathematics, took his first complex analysis course aged sixtysomething and was very surprised that every function was called $f$ and written as $f(z)$, even when considered as a function and not a specific value.)
(Real analysis probably has not experienced the same ossification since it's much more mainstream, has a pile of more recent work to tie directly into it, and was one of the main topics that the more rigorous mathematicians got hold of to spread their ideology: witness the horrors of the entirely symbolic definition of the limit.)
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.
The function $f$ takes $x$ to $f(x)$. When one writes $$f(x)=3x$$ they are not describing the function $3x$ but rather the function that takes an input and outputs three times said input.
It is often easiest to describe a function using notation like $f(x)$, For example, a function that takes an input and squares it can be understood through the expression $$f(x) = x^2$$ while an alternative is less intuitive: $f$ is a function that squares its input.
To quote my answer this question:
There is a technical difference between $f$ and $f(x)$ that I want to elaborate on.
$f$ is a function. $f(x)$ is a function evaluated at an indeterminate (this might be the wrong word for it, but my brain is on the fritz) point, $x$. Thus the sentence "$f(x)$ is differentiable" is technically incorrect because $f(x)$ is the value of $f$ at $x$ and is therefore a number and not a function. In contrast, "$f$ is differentiable" is correct (of course, its only correct when it is in fact true). For another example, "$f$ is given by $x^2+1$" is also wrong. $f$ is given by $\{(x,y)\in S:x^2+1=y\}$, or however your underlying foundation specifies functions. It is $f(x)$ that is given by $x^2+1$.
In practice, this difference rarely is relevant and the vast majority of people are content to wave their hands and ignore it. Instead, they use $f(x)$ to refer to both the function and the function evaluated at the indeterminate point.
Ironically, the main place that people attend to this difference is done incorrectly. People write $f\in\mathbb{Z}[x]$ thinking that that's preferred to $f(x)$, but $f$ is not an element of $\mathbb{Z}[x]$ because that's not a set of functions - it's a ring of numbers. It happens to be that these numbers have a particular correspondence to functions from $\mathbb{Z}$ to $\mathbb{Z}$ and that $\mathbb{Z}[x]$ is isomorphic to an interesting subset of that set of functions, but they are distinct structures.
As others pointed out, when writing $f(z)$ you basically are referencing "the value that $f$ takes when given an arbitrary value $z$", which can be anything.
My teachers used to say that whenever you use arbitrary terms, you must specify which values they can take, as in the example below :
$f(z) = z+4$ where $z \in \Bbb C$
Meaning "$f$ is a function that returns $z+4$ for any complex number $z$"
Amusingly, you can name either the function and the parameter whatever floats your boat. I knew someone who would draw skulls for functions and fruits for arbitrary params (and that bothered some teachers even though it was totally legit).
Still, to truly answer your question, if you want to stick to definitions, you have to distinguish the name of the function (which in your case is $f$ but could be anything) and the value it takes when given a certain value (what you wrote $f(z)$), and specify its domain ($\Bbb C$ in your case).
You can even have functions that take inputs in a domain and give results in another domain, as $$f : \Bbb R \to \Bbb C$$ $$x \to x + i$$ Read as "a function $f$ that takes a real input $x$ and returns a complex output $x+i$".
