About the fake abs : f(x) = f(-x)

[tommy1729]

When discussing fake function theory we came across the fake sqrt.

the fake abs function is then fake_abs(x) = fake_sqrt(x^2).

Clearly when we want a real-entire function f(x) to satisfy f(x) = f(-x) then we consider f(fake_abs(x)).

But what if we have a real-entire function f(x) and we want to remove the property f(x) = f(-x).

We can do many things like : g(x) = f(x) + exp(-x). (*)

But lets consider the context of fake functions :

Find g such that
f(x) = g(fake_abs(x))

=>

g(x) = f(fake_abs^[-1](x))

g(x) = f( sqrt ( inv_fake_sqrt(x) ) )

Lets call inv_fake_sqrt(x) := fakesquare(x).

g(x) = f ( sqrt ( fakesquare(x) ) )

Now since f(x) = f(-x) we have that F(x) = f(sqrt(x)) is also an entire function.

g(x) = F( fakesquare(x) )

Now if we want g(x) to be entire then F ( fakesquare(x) ) needs to be entire.

Since fakesquare is a multivalued function ( an inverse of an entire ) its not entire.

SO when is F( fakesquare(x) ) entire ?

And how does that look like ?

Also of interest ( when its not entire ) :

fake ( F ( fakesquare(x) ) )

From (*) one then also wonders about

F( fakesquare(x) ) - f(x)

and how that looks like.

I have some ideas and guesses but no evidence or plots.

Seems like chapter 2 in fake function theory.

The analogue questions exist for exp(x) instead of x^2 ;

removing the periodic property.

regards

tommy1729

[sheldonison]

When discussing fake function theory we came across the fake sqrt.
... fake_sqrt(x^2).

for large positive numbers,

this is true if |real(x)| is large enough, and |imag(x)| isn't too large
However, at the imaginary axis grows large exponentially, and does not behave like x at all. And g(x^2) never behaves like abs(x), anywhere in the complex plane

Here are some example calculations:
5^2, small error term
also a small error term, but not abs(x^2)
(5i)^2, huge error term, nowhere near 5i

[tommy1729]

Im aware that there is no fake abs(z) that is a good approximation on all of C.

That is why I used the letter x.

So fake in the sense of asymptotic near the real line , or fake of the absolute value of the reals if you like.

The focus for fake functions is Always mainly on the real line anyway , as you well know.

Im aware of what you said sheldon.

You are completely correct.

But I see no problem with that.

Or did you not claim a problem ?

Are you conjecturing a growth rate perhaps ?

regards

tommy1729