F ( 3 ) = 3 → 3 → 3 {\displaystyle F(3)=3\rightarrow 3\rightarrow 3} F ( 4 ) = 4 → 4 → 4 → 4 {\displaystyle F(4)=4\rightarrow 4\rightarrow 4\rightarrow 4} F ( n ) = n → n → ⋯ → n ⏟ n copies of n {\displaystyle F(n)={\begin{matrix}\underbrace {n\rightarrow n\rightarrow \cdots \rightarrow n} \\\ \ n{\mbox{ copies of }}n\end{matrix}}}
G ≡ { n → ⋯ ⋯ ⋯ ⋯ ⋯ → n ⏟ n → ⋯ ⋯ ⋯ ⋯ → n ⏟ ⋮ ⏟ n → ⋯ ⋯ → n ⏟ n → ⋯ → n ⏟ n copies of n } n → ⋯ ⋯ ⋯ ⋯ ⋯ → n ⏟ n → ⋯ ⋯ ⋯ ⋯ → n ⏟ G ≡ ⋮ ⏟ n → ⋯ ⋯ → n ⏟ n → ⋯ → n ⏟ n copies of n {\displaystyle 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 ( 3 ) = ( F ∘ F ∘ F ⏟ ) ( 3 ) 3 copies of F {\displaystyle H(3)={\begin{matrix}(\underbrace {F\circ F\circ F} )(3)\\3{\mbox{ copies of }}F\end{matrix}}} H ( 4 ) = ( F ∘ F ∘ F ∘ F ⏟ ) ( 4 ) 4 copies of F {\displaystyle H(4)={\begin{matrix}(\underbrace {F\circ F\circ F\circ F} )(4)\\4{\mbox{ copies of }}F\end{matrix}}} H ( n ) = ( F ∘ F ∘ ⋯ ∘ F ⏟ ) ( n ) n copies of F {\displaystyle H(n)={\begin{matrix}(\underbrace {F\circ F\circ \cdots \circ F} )(n)\\n{\mbox{ copies of }}F\end{matrix}}}
∫ e x = F ( u n ) {\displaystyle \int e^{x}=F(u^{n})}
