# Is there a multiple function composition operator? [duplicate]

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Is there a commonly-accepted operator which defines multiple function composition? I have not been able to find one on any of the related Wikipedia pages. In one of my proofs, I've been finding it very tiresome to continually write the following: $$f_n\circ f_{n-1}\circ\cdots\circ f_0$$ And I would really appreciate finding out if there is a common notation for what I'm trying to do. The above notation only works for a sequence of functions; I imagine there's probably a way of using this composition operator to compose over an arbitrary ordered set, sort of like $\bigcup_{i\in I}A_i$ allows a union over any (not necessarily countable) indexing set $I$, but without the ordering condition.

I'd use foldl1 (.), but that doesn't seem very appropriate for math homework.

I feel like I should add why I feel like one is necessary, on top of the additional expressiveness we gain from being able to compose over arbitrary ordered sets. The problem itself defines $F_n$ as the above repeated-composition function; indeed, we can define $F_n$ in a recursive manner with $F_0=f_0$ and $F_{n+1}=f_{n+1}\circ F_n$. However, the issue remains because in my proof I frequently need to expand $F_n$ (for instance, calculating $F_n'$), in which case such an operator would come in handy.

## marked as duplicate by Daniel FischerOct 24 '16 at 11:54

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## 3 Answers

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: $$\mathop{\bigcirc}^n_{k=0} f_k \quad \text{ or } \quad \mathop{\bigcirc}^0_{k=n} f_k \ .$$ 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).

• Thank you for the answer. I'm not $\LaTeX$-ing this assignment (for reasons that are not relevant). It seems that no such notation exists, then? I was hoping someone could have come across something in a paper they read before. – VF1 Jan 8 '15 at 23:33
• I haven't seen such a notation. But sometimes one sees things like $\displaystyle\prod_{k=1}^n\frac\partial{\partial x_k}$ for composition of differential operators. If I were writing something by hand I might just write something like "In the following the notation $\mathop{\bigcirc}^{n}_{k=0} f_k$ will mean $f_n\circ\cdots\circ f_0$." ${}\qquad{}$ – Michael Hardy Jan 8 '15 at 23:44
• I would be comfortable with that circle notation for functions which commute with each other. Otherwise, I would keep being confused which index was evaluated first! – Kyle Miller Oct 24 '16 at 1:15

I personally write $f_n\cdots f_0$ for the composition (possibly based on the analogy of multiplying matrices for composition of linear transformations).

Alternatively, you could let finite sequences stand for functions. For $F=\{f_i\}_{i=0}^n$ say $F$ is a function defined by $F(x)=f_n(\cdots f_1(f_0(x))\cdots)$.

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., $G_2 = f_0 \circ f_1 \circ f_2$.