User:TachyonJack/Sandbox 2

F
$$ F(3) = 3 \rightarrow 3 \rightarrow 3 $$

$$ F(4) = 4 \rightarrow 4 \rightarrow 4 \rightarrow 4 $$

$$ F(n) = \begin{matrix}

\underbrace{n\rightarrow n\rightarrow \cdots \rightarrow n}\\ \ \ n \mbox{ copies of }n \end{matrix} $$

$$ G \equiv \left.\{

\begin{matrix} &\underbrace{n\rightarrow \cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow \cdots \cdots\cdots \cdots \rightarrow n} \\ &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow \cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} \right \} \begin{matrix} &\underbrace{n\rightarrow \cdots \cdots\cdots\cdots\cdots \rightarrow n} \\ &\underbrace{n\rightarrow \cdots \cdots\cdots \cdots \rightarrow n} \\ G \equiv &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\ &\underbrace{n\rightarrow \cdots \cdots \rightarrow n} \\ &\underbrace{n\rightarrow \cdots \rightarrow n}\\ &n \mbox{ copies of }n

\end{matrix} $$

H
$$ H(3) = \begin{matrix}

(\underbrace{F \circ F \circ F})(3)\\ 3\mbox{ copies of } F \end{matrix} $$

$$ H(4) = \begin{matrix}

(\underbrace{F \circ F \circ F \circ F})(4)\\ 4\mbox{ copies of } F \end{matrix} $$

$$ H(n) = \begin{matrix}

(\underbrace{F \circ F \circ \cdots \circ F})(n)\\ n\mbox{ copies of } F \end{matrix} $$

OTHER
$$ \int e^x = F(u^n) $$