Complex Partway Functions

Dalek1099

What I mean by a partway function is this:
ff(x)=6x now as you probably know that f(x)=√6(x) or you could argue f(x)=-√6(x), with that function that you have just found being the partway function between x and f(x)=6x-Do you understand?
But what about more complex partway functions like ff(x)=sin(x) so what is f(x)= to?, which is the same as saying what is the partway function between x and sin(x).

Mark44

No. Please provide a definition for this term - partway function.

Mute

If I understand correctly, you are asking, if

$$f(f(x)) = \sin x,$$
then what is ##f(x)##?

This particular equation has been studied, but I'm afraid I can't remember the name of the function or the wikipedia article.

Edit: Ah! Here's the wikipedia article I was thinking of:

Schroder's equation

In particular, it discusses the "functional square root", a function such that ##h_{1/2}(h_{1/2}(x)) = h(x)##, which is relevant to the question of ##f(f(x)) = \sin x##.

Klungo

I would avoid using ff(x) to represent (f(x))^2 since it looks like the composite f(f(x)).

Also, I'm a bit confused about what you mean by complex (complex numbers or complicated).

That is, for y^2=6x, x>=0 if x is real or all x if we're using complex numbers.

In the case of y^2=sin(x), assuming we're working with real numbers, sin(x)>=0 so x€U(n)[2nπ,(2n+1)π] for n€Z.
So, y=+-√sin(x) for the same x in that set. (These are two distinct functions.)

If we work with complex functions, then y=+-√sin(x) for all complex numbers x.

Is this what you're looking for?

Dalek1099

You are on the right lines but your link doesn't include any information on that specific equation.

Mute

Yes, it doesn't discuss f(f(x)) = sin x in particular, but rather a more general problem. If you can solve the functional equation ##\Psi(\sin x) = s \Psi(x)## for ##\Psi(x)## (which you may not be able to do analytically), then the "half" sine function would be given by ##h_{1/2}(x) = \Psi^{-1}(s^{1/2} \Psi(x))##, such that ##h_{1/2}(h_{1/2}(x)) = \sin x##. By searching some papers about solving Schroder's equation perhaps you can find one which discusses the case of sin x.

Klungo

If you click on the link, "functional square root", you would find that rin(rin(x))=sin(x) where rin(x) is the function you supposedly wanted.

@dalek, in you first post where you said ff(x)=6x, and you also said f(x)=√6(x) which could mean either √(6x) or (√6)x. This was not clear as (f(x))^2 = 6x, and f(f(x))= 6x for the corresponding choice of f(x), respectively.

Dalek1099

Can you go through the notation because I don't understand it and is there a calculator on the internet for them, with degrees and radians?One of the reasons, I have asked this is that I have discovered some nice theorems to approximate such functions where x is in degrees,sin[2]x=sin sin x, sin[0.5]x roughly=(xsinx)^0.5, sin[0.25]roughly=x*(sinx/x)^0.25, sin[i]x roughly=x*(sinx/x)^i - you should be able to see a nice pattern going on there.

Dalek1099

Is there a rin(x) calculator on the internet?