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

[sheldonison]

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

\( f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)} \)
for large positive numbers, \( g(x)=f(x)\exp(-x) \approx \sqrt{x} \)

\( g(x^2) = g((-x)^2) \approx x\;\; \) this is true if |real(x)| is large enough, and |imag(x)| isn't too large
However, at the imaginary axis \( g(x^2) \) 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:
\( g(25)=5 + 1.5\cdot10^{-13}\;\;\; \) 5^2, small error term
\( g((5+0.1i)^2) = 5+0.1i - k\cdot10^{-13}\;\;\; \)also a small error term, but not abs(x^2)
\( g(-25)=-866955233 \;\;\; \) (5i)^2, huge error term, nowhere near 5i
\( g(25i) = 3.53768061172 + 3.52450328163i \;\;\;\;\sqrt{25i}\approx 3.535534 +3.535534i \)