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.
blaschkeProd(z, a)
| 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
|
The value of the Blaschke product at z.
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)).
Other maths:
jacobiTheta(),
juliaNormal(),
mandelbrot()
# 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 # }