I'm learning about the more powerful tab-completion and expansion capabilities of zsh, and they don't seem to work when I run zsh under emacs with M-x shell:

cat $PATH<TAB> expands the tab variable in Terminal, but in shell-mode it just beeps.

I poked around the emacs environment and here's what I found:

TAB (translated from ) runs the command completion-at-point, which is an interactive compiled Lisp function in `minibuffer.el'.

It is bound to TAB, .

(completion-at-point)

Perform completion on the text around point. The completion method is determined by `completion-at-point-functions'.

completion-at-point-functions is a variable defined in `minibuffer.el'. Its value is (tags-completion-at-point-function)

So I'm surmising I need to add a function to completion-at-point-functions, but which one?