Continued fraction expansion of the square-root operator
John C. Bancroft
A number of approaches to solving the wave equation require approximations to square-root of (1-x2). When x is very small, the Taylor series approximations are usually sufficient. When using the first two terms of the series, we get a parabolic equation that leads to the 15-degree approximation to the wave equation. Improved finite-difference solutions to very steep dips are found from rational functions that are often derived from continued fraction expansion.
A very simplified description of continued fraction expansion is presented, starting with real numbers and progressing to functions.