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