Continued Fractions. Volume 1: Convergence Theory by Lisa Lorentzen

By Lisa Lorentzen

Persisted Fractions includes volumes -- quantity 1: Convergence conception; and quantity 2: illustration of capabilities (tentative title), that's anticipated in 2011. quantity 1 is devoted to the convergence and computation of persevered fractions, whereas quantity 2 will deal with representations of meromorphic services through endured fractions. Taken jointly, the 2 volumes will current the elemental persisted fractions conception with out requiring an excessive amount of earlier wisdom; a few uncomplicated wisdom of complicated services will suffice. either new and complicated graduate scholars of endured fractions shall get a finished figuring out of ways those limitless constructions paintings in a few purposes, and why they paintings so good. A various buffet of attainable purposes to whet the urge for food is gifted first, ahead of the extra uncomplicated yet modernized idea is given. This new version is the results of an expanding curiosity in computing distinct features through persevered fractions. The equipment defined intimately are, in lots of instances, extremely simple, but trustworthy and effective. Contents: Introductory Examples; fundamentals; Convergence standards; Periodic and restrict Periodic persevered Fractions; Numerical Computation of persisted Fractions, a few endured Fraction Expansions.

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Extra resources for Continued Fractions. Volume 1: Convergence Theory

Example text

The figure shows the graph of F3 (z) for real z > 0. 1) is called stable if all zeros lie in the left half plane Re(r) < 0. The concept is important in the theory of linear differential equations with constant coefficients, as illustrated in the example below. Example 23. To solve the differential equation y¨ + 3y˙ + 2y = 0 where y˙ denotes the derivative with respect to the real variable t, one solves the corresponding characteristic equation Q2 (r) := r2 + 3r + 2 = 0. It has the roots r1 = −1 and r2 = −2.

The hypergeometric function 2 F1 (a, b; c; z) is the analytic function with power series expansion ∞ (a)n (b)n z n ab z a(a + 1)b(b + 1) z 2 =1+ + + ··· (c)n n! c 1! c(c + 1) 2! 2) at 0. Many useful special functions are special cases of 2 F1 (a, b; c; z). 2) is an infinite power series whose radius of convergence is 1. 2) by comparing the power series on both sides term by term: 2 F1 (a, b; c; z) = 2 F1 (a, b + 1; c + 1; z) a(c − b) z 2 F1 (a + 1, b + 1; c + 2; z) c(c + 1) 2 F1 (a, b + 1; c + 1; z) = 2 F1 (a + 1, b + 1; c + 2; z) − − (b + 1)(c − a + 1) z 2 F1 (a + 1, b + 2; c + 3; z).

1) For simplicity we assume that z = −1. Then its classical approximants fn (z) can be written f1 (z) = f2 (z) = f3 (z) = z(1 + z) z = , 1−z 1 − z2 z z(1 − z) z z(1 − z 2 ) = = , 1−z +1−z 1 − z + z2 1 + z3 z z z z(1 + z 3 ) , = 1−z +1−z +1−z 1 − z4 and by Problem 13 on page 49 with x = −z and y = 1, fn (z) = z(1 − (−z)n ) z + (−z)n+1 = . 1 − (−z)n+1 1 − (−z)n+1 We therefore distinguish between two cases: 0 < |z| < 1 : The continued fraction converges to z. Since fn (z) = z + (−z)n+1 + higher powers of z it corresponds at 0 to the series z + 0z 2 + 0z 3 + · · · |z| > 1 : The continued fraction converges to −1.

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