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#1 Re: Euler Avenue » x ° y = y ° x » Today 04:26:23

I should mention that the parametrisation
isn't the only choice we could have used in post #8 -- there are others we could use to generate solutions to your equation. (By the way, in mathematics we call this kind of 'guess' at a solution an ansatz. This is very common for solving things like difference equations where we don't know what the solution is, but we hazard a guess at what we think it 'should' look like -- and then prove it's a solution.) For example, let's consider something a bit more different. (This one is a lot more beautiful I think!)

As before, suppose that
satisfy
Let's instead consider a mapping
given by:


where
and
(These are called polar co-ordinates.) Then:

-- i.e. raising both sides to the power of

Expanding the brackets on both sides (because we want
on one side and
on the other), we get:

Pulling 'like' terms across the equality:

This gives us another tool for generating solutions in
and
since we can now just choose a particular
which will determine
and hence
and
For example, you can show that if we set

then it follows that
and hence

which is the classical solution to

In particular, we again see Bob's result showing up here too. Note that our expression for
in terms of
wasn't defined when
i.e. when
However, if we consider the limit instead, we get:

So as
then
and so we have:

i.e.
and
both converge to
as
approaches

By the way, the proof of the first line in this limit is as follows. Let's split the function up into two parts, i.e. let

so that

We'll try to calculate the limits of these functions as
separately. If both exist, then the product of these limits exists (and is equal to the limit of the product). We clearly have:

For the first function, we have:

The function inside the limit satisfies the conditions of L'Hopital's rule, so this is just equal to:

and so

#2 Re: Euler Avenue » x ° y = y ° x » Yesterday 23:05:46

You also mentioned the product-log function (also called the Lambert W-function). I'll try to explain where this comes from (and why WolframAlpha tends to spit up that kind of result!). First: what is it?

The definition of Lambert's W-function is that it's the inverse of this function below:

So in other words, if
then the W-function tells us which values of
we need, i.e.
OK great, but how does this relate to our equation? Let's go back to this line in post #8:

Now instead of putting
let's try a different substitution, say...
This gives us:

But from the definition of the W-function, the value of
which satisfies the above must be given by:

Re-expressing all of this in terms of
and
only, we get:

So there's another way we can generate some solutions. The problem is that the W-function isn't expressible (generally) in terms of elementary functions -- so given an arbitrary value of
your corresponding
which satisfies
might end up being (irreducibly) some infinite series!

The other issue is that I haven't defined the W-function particularly well. Does
even have an inverse? Well, no, not really -- as it stands, our W-function would be multi-valued, so that's no good. We instead need to do what's called a 'branch cut' in complex analysis. (This basically means you define the possible inputs for W such that it only spits out unique values.) So the full solution set would look something like:

where, for each
the function
represents a different branch of the W-function for each of the ranges of possible values of

I haven't done a full video on branch cuts but I do motivate them to some extent in this video if you want to have a look.

#3 Re: Euler Avenue » x ° y = y ° x » Yesterday 22:21:16

Relentless wrote:

Hi Bob,

I'm afraid I don't understand this result, where it comes from or how it is significant.

Bob's result can also be seen in another way: as can be seen from my previous post, one such 'solution set' is given by the following:



Clearly this doesn't make much sense when
We can, however, consider the limit as
so let's do that instead. Setting
  gives us:

One way of interpreting this result is that
is exactly the point at which the trivial solutions (that's everything on the line y = x) and the non-trivial solutions (that's the curved bit of Bob's graph) meet.

#4 Re: Euler Avenue » x ° y = y ° x » Yesterday 22:11:46

Hi Relentless,

I'm not sure what maths background you have but I'll try to explain what's going on as best I can -- please let me know if anything isn't clear (happy to talk through it in more detail if needed).

The standard method for generating such solutions is as follows. Consider the equation
Taking logs of both sides gives us:

i.e.

Now suppose that
for some value
This gives us:

i.e.

and since
then we also have

This is called a parametrisation. Now to generate some solutions to your equation we just need to vary
Clearly setting
gives us
so let's try a different value, say,
Simplifying, we get:


and similarly you can generate (infinitely) many similar solutions by plugging in some additional values of
(By the way -- these types of solutions are called 'algebraic numbers'.) OK, so how about negative solutions? Well, let's try setting
This gives us:


Note that the cube root of -2 has three solutions: one is real, the other are complex (and conjugate) roots -- so you could generate complex solutions this way, too.

#5 Re: Help Me ! » Proving that a function is surjective » 2021-09-23 09:31:31

Hi ziabing,

Welcome to the forum. I will assume here the natural numbers include 0 -- otherwise there is no
for which

To show that this function is surjective, you need to show that every element
can be 'hit' by some
under your function
. So first consider the case where
. Can you always find an
such that
?

Yes, because we can just take
in which case -- because
must be even --
And since
then it's clear that
and so it must be that

That takes care of the positive integers (and 0) in
-- now you just need to do the same for the negative integers in
and you're done. So let's consider the case where

So, given
can you find an
such that we'll always have
?

I will leave you to finish this off -- but please do post back if you need more help.

#6 Re: Maths Is Fun - Suggestions and Comments » New Members » 2021-08-11 11:21:49

Yes, most of these new members have unfortunately joined solely to advertise and we tend to remove them as soon as they post something. In the vast majority of cases these are fairly easy to spot as their e-mail addresses often show up on widely shared spam lists -- however, there are occasionally some that slip through the net particularly if they masquerade as genuine posters to begin with. We do try to deal with these at the earliest possible instance but naturally we cannot patrol the site every hour of the day (though we do our best!).

The forum's current infrastructure thankfully blocks most of this automatically -- for example, without the requirement to verify your e-mail address before posting we'd easily get over a hundred members flooding the forum within a day.

#9 Re: Help Me ! » Limit of Two-piece Function » 2021-06-05 22:48:01

Your conclusion is correct, but the same comments I made on your other piecewise function thread also apply here so I suggest you read that first (it's the same kind of set-up, just with two pieces rather than three).

The question did ask for you to use a graph. So what do you think the graph looks like and in particular what do you think it looks like near x = -1?

#10 Re: Help Me ! » Limit of Three-piece Function » 2021-06-05 22:42:33

Your conclusion is correct -- however, there are a few technical points to note.

nycmathguy wrote:

Find the limit of x^2 as x tends to 1 from the left side.

What you really want to be saying is that you are finding the limit of f(x) as x tends to 1 from the left, rather than the limit of x^2.

nycmathguy wrote:

(1)^2 = 1

Strictly speaking, this is how you evaluate the limit of f(x) as x approaches 1 from the left, but this isn't how you evaluate the left-hand limit of x^2, this is just substituting in x = 1 -- although you can get away with it here. (This is because x^2, 2 and -3x + 2 are all continuous functions, so the left and right-hand limits for each function exist, are equal, and are equivalent to their usual limits.) The way you ought to be thinking about it is:

It'll lead to the same answer, but conceptually this is along the sort of lines you'll be required to think through when you look at the epsilon-delta type questions.

However, the question did ask for you to use a graph. So what do you think f(x) might look like on a graph? What kind of shape does it have as you move from left to right? And in particular what happens near x = 1?

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