TITLE: simpler way to define modular forms
QUESTION [5 upvotes]: i'm trying to define modular forms for the full modular group for a general audience and want to make it as simple as possible. please tell me if this is correct:
definition of a modular form $f$:

holomorphic function $f$ from upper half plane to $\mathbb{C}$.
$f(z)$ bounded as Im($z$) tends to infinity.
$f$ satisfies the usual transformation law under $SL_2(Z)$

All the definitions I've seen say that, instead of 2., f should be holomorphic at infinity. I know that this is more beautiful and better for generalizations of the definition, for example to congruence subgroups... but then I have to explain what it means to be holomorphic at infinity, by using a power series representation. That's an extra step which I think is implicit in my definition because.....
By 3., $f(z)$ is periodic and so by complex analysis it must have a fourier series expansion $f(z)=\sum_{n=-\infty}^{\infty} a(n) e^{2\pi i nz}$. By 2. and complex analysis, we must have a(n)=0 for $n<0$. Then the resulting fourier series shows that $f$ is holomorphic at infinity and we arrive at the same thing.
Is all this right or am I missing something?

REPLY [11 votes]: This is not exactly an answer to the question as asked, but I was recently confronted to the same problem: how to define, for a general audience, a modular form for the full modular group
in a way which is both correct and quick? Here is a solution I used:
A modular form of weight $k$ (and level 1) is a formal series $f(q)=\sum_{n \geq 0} a_n q^n$, converging on the open unit disc, such that, in terms of the variable $z$ defined by $q=e^{2 i \pi z}$, $f$ satisfies $f(-1/z)=z^k f(z)$.
It is short and allows you to talk without further steps of the $q$-expansion of $f$, and avoid even mentioning the full modular group $Sl_2(\mathbb Z)$ or using any explicit element in it. This also avoids having to justify the existence of a $q$-expansion from $f(z+1)=f(z)$, for which I have always felt unsatisfied by the common three-word justification "by Fourier theory".  
Of course, depending of what you want to do next, this may be either an advantage or a shortcoming. If you need to discuss other congruences subgroups later, this is not the definition you want. But if what you need about modular forms is mainly their 
$q$-expansions, as in many arithmetical applications, this definition may be useful.