I'm currently writing my master thesis about the j-invariant and his q-expansion. 
Now i have the result that the growth of the coefficients is asymptotically 
$$c(n) \sim \frac{e^{4\pi \sqrt{n}}}{\sqrt{2}n^{\frac{3}{4}}}.$$


Is there any special reason to investigate this growth? Is it possible to use this result to make some statements about the j-invariant or is this just a "nice-to-know" information?


Thanks in advance. 



Asymptotic Formula of the coefficients of the q-expansion of the J-Invariant
There are lots of reasons why it's interesting to know the growth rate of modular functions. For the $j$-invariant, the coefficients are closely related to the dimensions of the irreducible representations of the monster group (the largest sporadic finite simple group), so the size of those coefficients tells you something about the representations of $M$. It's also not a coincidence that the growth rate of $c(n)$ looks a lot like the asymptotic formula for the partition function $p(n)\sim e^{\pi\sqrt{2n/3}}/4\sqrt3 n$, and the growth rate of $p(n)$ is of interest in many sorts of combinatorial calculations.


