09/16/2014, 12:27 PM
Ok my next idea.
Im having ideas to prove the validity of the Cauchy integral for fake function theory.
Some conditions first :
Lets call our real-analytic function f(z) of which we want a +fake.
1) there needs to be an annulus around the origin that contains at most one branch.
2) the Riemann surface needs to be "well-connected".
As example : log(z^3) log(z^5) is not well connected.
Plot it near the origin to see it.
3) f(z) has no essential singularity.
4) f(x) , f ' (x) , f " (x) > 0 for x > 0
Now we use an old idea of me
f(z) = +Taylor_1(z) + +Taylor_2(z/(z+a_1)) + +Taylor_3(z/(z+a_2))
where +Taylor means a Taylor series with positive real coefficients.
a_1,a_2 are selected positive reals.
There series expansions MUST have a +fake described by the Cauchy.
to be continued.
regards
tommy1729
Im having ideas to prove the validity of the Cauchy integral for fake function theory.
Some conditions first :
Lets call our real-analytic function f(z) of which we want a +fake.
1) there needs to be an annulus around the origin that contains at most one branch.
2) the Riemann surface needs to be "well-connected".
As example : log(z^3) log(z^5) is not well connected.
Plot it near the origin to see it.
3) f(z) has no essential singularity.
4) f(x) , f ' (x) , f " (x) > 0 for x > 0
Now we use an old idea of me
f(z) = +Taylor_1(z) + +Taylor_2(z/(z+a_1)) + +Taylor_3(z/(z+a_2))
where +Taylor means a Taylor series with positive real coefficients.
a_1,a_2 are selected positive reals.
There series expansions MUST have a +fake described by the Cauchy.
to be continued.
regards
tommy1729
