09/01/2014, 10:24 PM
It seems my friend ( our ? ) mick has given us ( myself and sheldon in particular ) credit for the use of " fake function theory " 
here is the link :
http://math.stackexchange.com/questions/...for-real-x
As I mentioned before z - ln(2sinh(z)/z) is asymptotic to ln(z) near the real line, but it fails to be entire ( sinh is periodic ! ).
Guess we could consider the fake ln(x^2 + 1) here too.
Unlike the half-exp we are considering more standard functions which might give intresting closed form results !
Since he asked for integral representations this crossed my mind :
fakeln(z) = integral dz / fakesqrt(1+z^2)
thereby changing the idea of a fake log to a fake sqrt such that
fakesqrt(1+z^2) =/= 0 for any z.
That implies ( i think ? ) ln(fakesqrt(z)) = entire ?
hmmm.
I wrote about fakesqrt once ... (thinking)
regards
tommy1729
here is the link :
http://math.stackexchange.com/questions/...for-real-x
As I mentioned before z - ln(2sinh(z)/z) is asymptotic to ln(z) near the real line, but it fails to be entire ( sinh is periodic ! ).
Guess we could consider the fake ln(x^2 + 1) here too.
Unlike the half-exp we are considering more standard functions which might give intresting closed form results !
Since he asked for integral representations this crossed my mind :
fakeln(z) = integral dz / fakesqrt(1+z^2)
thereby changing the idea of a fake log to a fake sqrt such that
fakesqrt(1+z^2) =/= 0 for any z.
That implies ( i think ? ) ln(fakesqrt(z)) = entire ?
hmmm.
I wrote about fakesqrt once ... (thinking)
regards
tommy1729
