Calculations of the Invariant Measure for Hurwitz Continued Fractions

Measure theory and Perron-Frobenius operators for continued fractions

The Hurwitz Complex Continued Fraction

The Hurwitz complex continued fraction algorithm generates Gaussian rational approximations to an arbitrary complex number α by way of a sequence (a0, a1, . . .) of Gaussian integers determined by a0 = [α], z0 = α − a0, (where [u] denotes the Gaussian integer nearest u) and for j ≥ 1, aj = [1/zj−1], zj = 1/zj−1−aj . The rational approximations are the finite continued fractions [a0; a1, . . . ,...

Generating bessel functions in mie scattering calculations using continued fractions.

A new method of generating the Bessel functions and ratios of Bessel functions necessary for Mie calculations is presented. Accuracy is improved while eliminating the need for extended precision word lengths or large storage capability. The algorithm uses a new technique of evaluating continued fractions that starts at the beginning rather than the tail and has a built-in error check. The conti...

Generalized Continued Logarithms and Related Continued Fractions

We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have fin...

Continued Logarithms and Associated Continued Fractions

We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation we discover that, for one such for...