# Notation for repeated composition of functions

I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ \cdots \circ {\tau _n}(z)$

By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n {} ,\bigcup\limits_{i = 1}^n {} ,\bigcap\limits_{i = 1}^n {} ,$ I want to write ${T_n}(z) = \left( {\mathop \circ \limits_{i = 0}^n {\tau _i}} \right)(z)$ or even ${T_n}(z) = {\mathop \circ \limits_{i = 0}^n {\tau _i}} (z)$. Can I do this?

Basically you can do anything. Notice, however, that the classical operators $\sum$ and $\prod$ do not coincide with the symbols they represent. And old books used to have $\sum$ instead of $\bigcup$ and $\prod$ instead of $\bigcap$.

I personally understand the notation $\mathop{\circ}_{n=1}^N$, but it doesn't look appealing. By analogy, why don't you define $$\mathop{\rm C}\limits_{n=1}^N$$ or $$\mathop{\rm K}\limits_{n=1}^N ?$$ Anyway, I don't like them either...

• I can't use K as that's already in use for continued fractions, because K = Kettenbruch, the German for Continued Fraction. – BudgieJane Sep 10 '14 at 17:55

If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do: $$\overset{n}{\underset{k=0}\bigcirc}\ f_k \quad \text{ or } \quad \overset{0}{\underset{k=n}\bigcirc} f_k \ .$$ Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $$\displaystyle\bigcap$$ in $$\displaystyle\bigcap_{k=0}^n A_k$$ or to $$\displaystyle\bigoplus$$. Before the \begin{document} I'd put \newcommand{\Circ}{blah blah blah} (with a capital "C" distinguishing it from \circ.

Let $f_n(x)$ be a sequence of functions indexed by $n$.

Define a new sequence of functions $F_k(x)$ indexed by $k$:

$$F_k(x) = \begin{cases} f_0(x) &: k=0\\ (f_k\circ F_{k-1})(x)&: k\gt 0 \end{cases}$$

So you have, for example \begin{align} F_0(x) &= f_0(x)\\ F_1(x) &= (f_1 \circ f_0)(x)\\ F_2(x) &= (f_2 \circ F_1)(x) = (f_2 \circ f_1 \circ f_0)(x)\\ &\dots\\ F_{n}(x) &= (f_n\circ f_{n-1} \circ \cdots \circ f_0)(x)\\ \end{align}

Then for your composition $f_n\circ f_{n-1} \circ \cdots \circ f_0$ of $n$ terms you can simply write $F_n$.

Of course you can define a new sequence $G_n$ for the ascending direction, i.e., $G_2 = f_0 \circ f_1 \circ f_2$.