@Jimster41 - ##\mathbb{Z}## is the symbol for the set of integers. However, when working with power series we are generally summing complex numbers, and it is customary to use lower case ##z## to denote a general complex number. The reason it's useful to talk about power series in this context is because we have straightforward convergence theorems for them, and because a Fourier series of the form
$$\sum_{n=0}^{\infty} a_n e^{i n x}$$
is simply the power series
$$\sum_{n=0}^{\infty}a_n z^n$$
evaluated at ##z = e^{ix}##, which is simply the unit circle since ##|e^{ix}| = 1##.
"Sup" means supremum, which is the same as "max" when working with finite sets, but it also generalizes to infinite sets which may not have a maximum value. For example, the interval ##[0,1)##, which is the set of all real numbers ##x## satisfying ##0 \leq x < 1##, has no maximum value, but its supremum equals ##1##.
"Lim sup" means "limit superior", and it is a notion associated with sequences of real numbers. It is the unique number ##L## such that only finitely many of the terms of the sequence exceed ##L##, but infinitely many terms exceed any number smaller than ##L##. So for example, if ##a_n = (-1)^n##, the terms oscillate between ##1## and ##-1##. The lim sup of this sequence is ##1##. And the lim inf (which is defined analogously to lim sup) is ##-1##. If the sequence has a limit, then the lim sup and lim inf are both equal to the limit.
Monstrous moonshine is related to the monster group from group theory, and is completely unrelated to this function. I haven't heard this function called the Weierstrass Monster Function until now, but I like it!
P.S. If you are reading this directly below the blog post and the typesetting comes out garbled, try reading it in the thread associated with the blog post:
https://www.physicsforums.com/threa...entiable-function-part-1.811795/#post-5096693 It's exactly the same post, but for some reason it is not displaying correctly when viewed on the blog page.
