06/03/2014, 12:16 PM
Somewhat of a followup :
We can say D(z + theta_A(z)) = z + theta_B(z).
Then by substitution u = z + theta_A(z) , z + theta_A(z) = u + theta_C(u) we get :
D(u) - u = theta_C(u)
rewrite u as z we get :
D(z) - z = theta(z)
Now this is a yes/no question. IS D(z) - z a 1periodic holomorphic function that satisfies the 5 conditions or not ?
If the answer is yes , we have existance and a way to compute.
Not sure about uniqueness. It seems we do not have uniqueness but we can make it unique by adding another theta shift and require local boundedness.
If the answer is no then .... euh ... I dont know what to conclude.
... If we weaken the 5 conditions then the steps are no longer necc valid.
Assuming yes , for analysis it seems intesting to investigate the extrema of D(z) - z , if we find a single max and no min we can use carleman matrices to solve the equation.
Hence D ' (z) - 1 = 0 is an intresting equation.
Cauchy can then be used too btw.
Still thinking ...
regards
tommy1729
We can say D(z + theta_A(z)) = z + theta_B(z).
Then by substitution u = z + theta_A(z) , z + theta_A(z) = u + theta_C(u) we get :
D(u) - u = theta_C(u)
rewrite u as z we get :
D(z) - z = theta(z)
Now this is a yes/no question. IS D(z) - z a 1periodic holomorphic function that satisfies the 5 conditions or not ?
If the answer is yes , we have existance and a way to compute.
Not sure about uniqueness. It seems we do not have uniqueness but we can make it unique by adding another theta shift and require local boundedness.
If the answer is no then .... euh ... I dont know what to conclude.
... If we weaken the 5 conditions then the steps are no longer necc valid.
Assuming yes , for analysis it seems intesting to investigate the extrema of D(z) - z , if we find a single max and no min we can use carleman matrices to solve the equation.
Hence D ' (z) - 1 = 0 is an intresting equation.
Cauchy can then be used too btw.
Still thinking ...
regards
tommy1729