# Summation sign for composition

1. Oct 15, 2015

### disregardthat

I was wondering if anyone knew the standard notation for the following. Suppose I have functions $f_1,f_2 \ldots,f_n$, is there a compact way of writing $f_1 \circ f_2 \circ \ldots \circ f_n$ ? I am debating whether $\bigcirc^n_{i=1} f_i$ is proper or good notation. Have anyone encountered this notation?

2. Oct 15, 2015

### Staff: Mentor

This seems fairly compact to me.
IMO, no. I have never seen this used as notation (which doesn't mean that no one has ever used it). I recommend sticking with $f_1 \circ f_2 \circ \ldots \circ f_n$, as its meaning is obvious while your $\bigcirc$ notation is not.

3. Oct 15, 2015

### disregardthat

In my case, each $f_i$ is a series of compositions $g_{i1} \circ g_{i2} \ldots \circ g_{ir_i}$ as well, (in fact, each $g_{ij}$ is equipped with detailed superscripts). Furthermore, it is not one such $f_1 \circ \ldots f_n$ I want to write out, it is a composition of three such expressions, $f_1 \circ \ldots \circ f_n$, $h_1 \circ \ldots \circ h_m$, and $t_1 \circ \ldots \circ t_k$, each $f_i, h_i$ and $t_i$ on the form above, each with detailed superscripts. Writing it out in full will take too much space and look too cluttery.

4. Oct 15, 2015

### Staff: Mentor

How about this? $f_1 \circ f_2 \circ \ldots \circ f_n$, where $f_i = g_{i1} \circ g_{i2} \circ \ldots \circ g_{ik}$

5. Oct 15, 2015

### disregardthat

That is what I wanted to avoid. Just like $\sum_{ij} a_{ij}$ is neater than "$b_{1} + b_{2} + \ldots$, where $b_i = a_{i1} + a_{i2} + \ldots$"

6. Oct 15, 2015

### Fredrik

Staff Emeritus
I haven't seen it.

It's not obvious what $\bigcirc_{i=1}^3 f_i$ should mean.

\begin{align*}
&f_1\circ(f_2\circ f_3)\\
&(f_1\circ f_2)\circ f_3\\
&f_3\circ(f_2\circ f_1)\\
&(f_3\circ f_2)\circ f_1
\end{align*} When you're dealing with linear operators, the notation $AB$ is preferred over $A\circ B$, and you could write $\prod_{i=1}^n A_i$.

7. Oct 15, 2015

### aikismos

Obviously, if you clearly define it and use it repeatedly, readers will be able to cope, but stylistically, it comes across as a circle as used in geometry. Why not define a functional notation using subscript? $C(f)_{1,n} := f_1 \circ f_2 \circ \ldots \circ f_n$ While personally I like the idea of having a consensually acceptable operator in symmetry to sigma notation, you probably are better off sticking with something a little more conventional.

\$0.02,
jtv