Where is the Purrr ~ operator documented?

When you use ~ within the context of purrr functions, it will be passed to the as_mapper() function, which in turn passes on to the as_function() function from rlang. These help files have a very basic bit of what is needed to use this. This is further documented in the Advanced R Book Chapter 9, Section 9.22, which has a few good examples, and this chapter goes on to continue those ideas.

There is a good explanation given AdvanceR (link given in another answer). There is also a short description (usage example) given in purrr cheatsheat first page bottom left.

The usage of multiple arguments with twiddle ~ may be seen at with documentation of purrr given in its different functions. e.g. map see argument description which states that

.f A function, formula, or vector (not necessarily atomic). If a function, it is used as is. If a formula, e.g. ~ .x + 2, it is converted to a function. There are three ways to refer to the arguments: For a single argument function, use .

For a two argument function, use .x and .y

For more arguments, use ..1, ..2, ..3 etc

This syntax allows you to create very compact anonymous functions.

Moreover, R in its newest version (4.1.0) has also started similar kind of shorthand notation of functions

R now provides a shorthand notation for creating functions, e.g. \(x) x + 1 is parsed as function(x) x + 1.

This shorthand notation may also provide useful in functions outside tidyverse, with only differentiation from twiddle being that here arguments are not named by default. But again, this non-default naming may also be proved useful when one invisible function is to be used inside another and twiddle style of notation will not work in that case.