blaschkeProd: Calculate Blaschke products

Description

This function calculates Blaschke products (https://en.wikipedia.org/wiki/Blaschke_product) for a complex number z given a sequence a of complex numbers inside the unit disk, which are the zeroes of the Blaschke product.

Usage

blaschkeProd(z, a)

Value

The value of the Blaschke product at z.

Arguments

z: Complex number; the point in the complex plane to which the output of the function is mapped
a: Vector of complex numbers located inside the unit disk. At each a, the Blaschke product will have a zero.

Details

A sequence of points a[n] located inside the unit disk satisfies the Blaschke condition, if sum[1:n] (1 - abs(a[n])) < Inf. For each element a != 0 of such a sequence, B(a, z) = abs(a)/a * (a - z)/(1 - conj(a) * z) can be calculated. For a = 0, B(a, z) = z. The Blaschke product B(z) results as B(z) = prod[1:n] (B(a[n], z)).

Examples

Run this code

# Generate random vector of 17 zeroes inside the unit disk
n <- 17
a <- complex(modulus = runif(n, 0, 1), argument = runif(n, 0, 2*pi))
# \donttest{
# Portrait the Blaschke product
phasePortrait(blaschkeProd, moreArgs = list(a = a),
  xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
  nCores = 1) # Max. two cores on CRAN, not a limit for your use
  # \dontshow{
  # R CMD check: make sure any open connections are closed afterward
  foreach::registerDoSEQ()
  doParallel::stopImplicitCluster()
  # }
# }