An implementation of the product formula for the Schottky-Klein prime function.
This function computes a truncated half Schottky group necessary to calculate the truncated product formula for the Schottky-Klein prime function. It returns a function handle which evaluates the prime function. It's main interest is as a check against other code which calculates the prime function.
Input:
- dv = a vector of circle centers.
- qv = a vector of circle radii.
- L = (optional) truncation level of the product formula (default L=4).
Output:
- wf = a function handle to the prime function with signature
w = wf(z, alpha), where z is an array of points at which to evaluate the function, and alpha is a scalar parameter value.
Example:
>> dv = [0.5, 0.5i];
>> qv = [0.1, 0.1];
>> wf = skprod(dv, qv, 6);
>> w = wf(-0.5-0.5i, 1);
>> w^2
ans =
2.39754812001042 + 1.76164377385124i
This value may be checked against
- D. G. Crowdy and J. S. Marshall, "Computing the Schottky-Klein prime function on the Schottky double of planar domains," CMFT 7 (2007) no. 1, 293-308.
The prime function is documented in
- H. Baker, Abelian Functions and the Allied Theory of Theta Functions, Cambridge University Press, Cambridge, 1897, 1995.