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 Mathematicae Varietas

7

Analytic Theory of Continued Fractions

As an example of expansion into a CF of an analytic function, consider:  F(z) = Arctan(z) .

From  Continued Fractions: Analytic Theory and Applications, by W. Jones and W. Thron (1980), we have the following (written using a common format for CFs):

Valid nearly everywhere in the complex plane as the single-valued branch of the analytic function with branch points at  z = i and z = -i, and branch lines north of  i ,  and south of   -i .  To compute  π/4 = arctan (1) to seven decimal places requires merely going out to the 9th convergent of the CF. Whereas, using the standard power series expansion:

Valid only within the unit disc, one needs to employ (approximately) the first 500,000 terms of the series. Thus this example demonstrates the two previously stated reasons for using Cfs as functional expansions: greater speed of convergence  and  enhanced region of convergence.

The speed of convergence of the CF expansion of Arctan(z), showing results for the 15th convergent vs the 30th convergent. Graph is centered at the origin, extending 10 units right and left ; branch lines above z = i  and below  z = -i   are clearly shown. The convergent is, of course, a rational function approximation to the Arctangent, and poles are illuminated along the branch lines. Dark areas indicate rapid convergence of the CF.

A CF expansion of the function   F(z) = - Ln( 1 - z )

,

converges and represents a single-valued branch of the function in the complex plane, with branch cut along the real axis to the right of   z = 1.  The speed of convergence is displayed in the following graphic, where we compare the 10th convergent to the 30th. As usual, dark = rapid, light = slow. Since any convergent is merely a rational approximation to the function, poles are seen illuminated along the branch cut. The scope of the figure is |x|, |y| < 10.

As we have seen before, the power series expansion of this function about  z = 0  fails to even converge outside of the unit disc.