anyways filog is y=x o y where o is a operator
for the case of normal filog it is ^
to form filog of an arbutary operator
computationally you can calculate ( x o y ) - y = 0 using newtons method
mathematicly youll need inverses
with the log-like inverse you can get
log_x(y)=y which is generally a stump
if a o b can be decomposed into f(a) ö g(b) we can go further (example x * cos (y) )
let inverse of ö to be õ (log like) and õ' is the other one
then we can get 1 õ f(x) = g(y) õ' y
from here you need a special inverse function (in case of x cos y it is inverse of cos(y)/y )
else if a o b can be decomposed into f(f*(x) ö y) we cen do other stuff
f* is log like inverse
y = f(f*(x) ö y)
f* (y) = f* (x) ö y
õ is iverse of ö of idk which side
f* (y) õ y = f*(x)
f(f*(y) õ y) = x
x = f(f*(y) õ y)
which we can use newton method / inverse from here but there is a better way if we have a inverse of õ such that a õ (t(b)) = a ö b (reciprical when ö is * and õ is /)
x = f(f*(y) ö t(y))
f* (x) = f*(y) ö t (y)
f* (x) = f*(y) õ y
now fun stuff let r(y) be t(f(a)) = f(r(y)) (r is - in normal filog)
f* (x) = f* (f(f* (y)) õ (f(f*(y))
f* (x) = f* (f(f* (y)) ö (f(r(f*(y)))
r(f* (x)) = r(f* (f(f* (y)) ö (f(r(f*(y))))
now we have a product where we can product log
let w be inverse of b ö f(b)
w(r(f* (x)) = r(f* (y))
r(w(r(f* (x))) = f* (y)
f(r(w(r(f*(x)))) = y
let F be f(r(w(r(f*(x)))) for ease
refundemantaling this will be hard but lets go
f(r(x)) = t(f(a)) by definition
F = t(f(w(r(f* (x))))
r(x)= f* (t(f(a))
F = t(f(w(f* (t(f((f* (x))))))
F = t(f(w(f* (t(x)))))
or alternatively
F = f(r(w(f* (t(x)))))
lets not know j
f* (t (x)) = j
t(x) = f(j)
t(t(x)) = x for further simplification (you can proof using the inverse of inverse = identity)
x = t(f(j))
x = f(r(j))
f* (x) = r(j)
r(r(x)) = x too bc if t(t(x))=x
r(f*(x) = j
f* (t (x)) = r(f*(x))
conclusively
F is filog
o is ^
f is exp
ö is *
õ is /
w is lambertw
t is 1/x
r is -x
f* is ln
or
F is filog
o : input operator
f,ö : x o y = f(f*(x) ö y)
õ : x ö y = z -> z õ x = y
w : x ö f(x) = y -> x = w (y)
t : x õ (t(y)) = x ö y
r : r(x) = f* (t(f(a))
f* : f(x) = y -> x = f* (y)
F = t(f(w(f* (t(x)))))
or alternatively
F = f(r(w(r(f* (x)))))