About the fake abs : f(x) = f(-x) - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: About the fake abs : f(x) = f(-x) ( /showthread.php?tid=942) |

About the fake abs : f(x) = f(-x) - tommy1729 - 12/11/2014
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 RE: About the fake abs : f(x) = f(-x) - sheldonison - 12/11/2014
(12/11/2014, 01:22 PM)tommy1729 Wrote: When discussing fake function theory we came across the fake sqrt. 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 RE: About the fake abs : f(x) = f(-x) - tommy1729 - 12/11/2014
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 |