Continued Fractions For Arctangent
Two alternatives to the power series for arctangent are these generalized continued fractions:
The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.
Read more about this topic: Inverse Trigonometric Functions
Famous quotes containing the word continued:
“The protection of a ten-year-old girl from her father’s advances is a necessary condition of social order, but the protection of the father from temptation is a necessary condition of his continued social adjustment. The protections that are built up in the child against desire for the parent become the essential counterpart to the attitudes in the parent that protect the child.”
—Margaret Mead (1901–1978)