2020-08-05 18:59:01 UTC, rjf:
>
> There are two square roots. In this (classic) integration
> example/bug, a choice has to be made. You know that 4 has
> two square roots, -2 and 2. The integrand, which also can
> be rewritten as sqrt ( 4-4*cos(x/2)^2) , has 2 square
> roots. Therefore there are two potential different values
> for the integral. Any answer that supplies only one answer
> is wrong.
Would you agree that:
- There is a map f from the real interval [0, +oo)
to itself, mapping each element to its square.
- That map is a continuous increasing bijection.
- It has a compositional inverse F which is also
a continuous increasing bijection
from the real interval [0, +oo) to itself.
The map f is commonly called "square" and the map F
is commonly called "square root". The notation
$x \mapsto x^2$ is often used for f, while the
"radical" notation $x \mapsto \sqrt{x}$ is often
used for F. In computer algebra systems, the name
`sqrt` is frequently used for F, while f rarely gets
named at all, with `sq`, `sqr`, `square` possibly
infrequently used.
You seem to be objecting strongly (please confirm)
about one or several of the following:
- the naming "square" for f or "square root" for F
- the "radical" notation $\sqrt$ for F
- using `sqrt` for F in computer algebra systems
If so, do you have any suggestion for
- admissible ways to name them?
- admissible mathematical notation?
- admissible naming in computer algebra systems?
These maps are sometimes useful to consider.
Computing their integrals along some segments
of the interval [0, +oo) is sometimes needed.
Is it admissible to compute such integrals?
How can we talk about these computations?
For sure,
- There is also a map g from the interval (-oo, +oo)
to the interval [0, +oo), mapping each element
to its square;
this map is surjective; under it, 0 has a single
preimage while all other real numbers have two;
it is therefore not bijective; the inverse relation
"sends" 0 to itself and any positive real y
to the two opposite reals whose square y is.
- There is also a map h from the field of complex numbers
to itself, mapping each element to its square;
this map is surjective; under it, 0 has a single
preimage while all other complex numbers have two;
it is therefore not bijective; the inverse relation
"sends" 0 to itself and any nonzero complex number z
to the two opposite complex numbers whose square z is.
I have observed that frequently, discussing the map F
or the computation of its integral along a subinterval
of [0, +oo) leads to lengthy discussions about the map g
or the map h and their inverse relations.
I am wondering about any appropriate vocabulary that can
be used in order to discuss the maps f and F without having
to discuss the maps g and h and their inverse relations.
The maps g and h and their inverse relations are important.
It is sometimes important not to forget that they are not
bijective and that most elements in their range have two
preimages under them. But sometimes that's not the topic,
and people are just discussing the map F.
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