Continued Fractions with Applications

Contents......Page all_32576_to_00623.cpc0009.djvu
I Introductory examples......Page all_32576_to_00623.cpc0017.djvu
1.1 Prelude to a definition......Page all_32576_to_00623.cpc0019.djvu
1.2 Formal definition. Convergence. Notation......Page all_32576_to_00623.cpc0023.djvu
2.1 The very best......Page all_32576_to_00623.cpc0026.djvu
2.2 A differential equation......Page all_32576_to_00623.cpc0029.djvu
2.3 An expansion of a function......Page all_32576_to_00623.cpc0030.djvu
2.4 A log-expansion......Page all_32576_to_00623.cpc0033.djvu
3.1 Hypergeometric functions......Page all_32576_to_00623.cpc0034.djvu
3.2 From power series to continued fractions......Page all_32576_to_00623.cpc0037.djvu
3.3 From continued fractions to power series......Page all_32576_to_00623.cpc0038.djvu
3.4 One fraction, two series......Page all_32576_to_00623.cpc0039.djvu
3.5 The length of an elliptic orbit......Page all_32576_to_00623.cpc0042.djvu
4.1 Śleszyński-Pringsheim's Theorem......Page all_32576_to_00623.cpc0046.djvu
4.2 Van Vleck's Theorem......Page all_32576_to_00623.cpc0048.djvu
4.3 Worpitzky's Theorem......Page all_32576_to_00623.cpc0051.djvu
5.1 Critical remarks on convergence......Page all_32576_to_00623.cpc0053.djvu
5.2 Modified approximants......Page all_32576_to_00623.cpc0054.djvu
5.3 Another concept of convergence......Page all_32576_to_00623.cpc0057.djvu
5.5 Computation of approximants......Page all_32576_to_00623.cpc0060.djvu
Problems......Page all_32576_to_00623.cpc0062.djvu
Remarks......Page all_32576_to_00623.cpc0066.djvu
References......Page all_32576_to_00623.cpc0068.djvu
II More basics......Page all_32576_to_00623.cpc0071.djvu
1.1 Tails......Page all_32576_to_00623.cpc0072.djvu
1.2 Tail sequences......Page all_32576_to_00623.cpc0075.djvu
1.3 Some properties of linear fractional transformations......Page all_32576_to_00623.cpc0078.djvu
1.4 Speed of convergence. Truncation error bounds......Page all_32576_to_00623.cpc0079.djvu
1.5 More about general convergence......Page all_32576_to_00623.cpc0082.djvu
2.1 Generating a continued fraction from a sequence......Page all_32576_to_00623.cpc0085.djvu
2.2 Equivalence transformations......Page all_32576_to_00623.cpc0088.djvu
2.3 The Bauer-Muir transformation......Page all_32576_to_00623.cpc0092.djvu
2.4 Contractions and extensions......Page all_32576_to_00623.cpc0099.djvu
Problems......Page all_32576_to_00623.cpc0102.djvu
References......Page all_32576_to_00623.cpc0107.djvu
III Convergence criteria......Page all_32576_to_00623.cpc0109.djvu
1.1 The Stern-Stolz divergence theorem......Page all_32576_to_00623.cpc0110.djvu
1.2 Continued fractions with positive elements......Page all_32576_to_00623.cpc0112.djvu
2.2 Classification of linear fractional transformations......Page all_32576_to_00623.cpc0117.djvu
2.3 Convergence of periodic continued fractions......Page all_32576_to_00623.cpc0120.djvu
2.4 Thiele oscillation......Page all_32576_to_00623.cpc0121.djvu
2.5 Tail sequences......Page all_32576_to_00623.cpc0122.djvu
3.1 Convergence sets......Page all_32576_to_00623.cpc0124.djvu
3.2 Value sets......Page all_32576_to_00623.cpc0126.djvu
3.3 Value set techniques I. A posteriori truncation error bounds......Page all_32576_to_00623.cpc0130.djvu
3.4 Value set techniques II. A priori truncation error bounds......Page all_32576_to_00623.cpc0132.djvu
3.5 Value set techniques III. The Hillam-Thron theorem......Page all_32576_to_00623.cpc0135.djvu
3.6 Value set techniques IV. The Stieltjes-Vitali theorem......Page all_32576_to_00623.cpc0139.djvu
3.7 Smaller value sets for truncation error bounds......Page all_32576_to_00623.cpc0141.djvu
4.1 Two useful lemmas......Page all_32576_to_00623.cpc0142.djvu
4.2 Parabola Theorems......Page all_32576_to_00623.cpc0146.djvu
4.3 S-fractions......Page all_32576_to_00623.cpc0154.djvu
4.4 Oval theorems......Page all_32576_to_00623.cpc0157.djvu
5.2 Finite limits, loxodromic case......Page all_32576_to_00623.cpc0166.djvu
5.3 Finite limits, parabolic case......Page all_32576_to_00623.cpc0173.djvu
5.4 Finite limits, elliptic case......Page all_32576_to_00623.cpc0175.djvu
5.5 Choice of approximants......Page all_32576_to_00623.cpc0176.djvu
5.6 Continued fractions K(a_n/1) where a_n \rightarrow \infty......Page all_32576_to_00623.cpc0185.djvu
5.7 Analytic continuation......Page all_32576_to_00623.cpc0190.djvu
Problems......Page all_32576_to_00623.cpc0193.djvu
Remarks......Page all_32576_to_00623.cpc0198.djvu
References......Page all_32576_to_00623.cpc0200.djvu
IV Continued fractions and three-term recurrence relations......Page all_32576_to_00623.cpc0205.djvu
1.1 The structure of the solution space......Page all_32576_to_00623.cpc0207.djvu
1.2 Approximants for periodic continued fractions in closed form......Page all_32576_to_00623.cpc0210.djvu
1.3 Linear independence of two solutions......Page all_32576_to_00623.cpc0212.djvu
1.4 The adjoint recurrence relation......Page all_32576_to_00623.cpc0213.djvu
1.5 Recurrence relations in a field F......Page all_32576_to_00623.cpc0216.djvu
2.1 Pincherle's theorem......Page all_32576_to_00623.cpc0217.djvu
2.2 Auric's theorem......Page all_32576_to_00623.cpc0222.djvu
3.1 Connection to recurrence relations......Page all_32576_to_00623.cpc0225.djvu
3.2 Minimal solutions and value sets......Page all_32576_to_00623.cpc0227.djvu
3.3 Tails and convergence......Page all_32576_to_00623.cpc0228.djvu
4.1 Forward stability of recurrence relations......Page all_32576_to_00623.cpc0234.djvu
4.2 A method for computing minimal solutions......Page all_32576_to_00623.cpc0236.djvu
5.1 Introduction......Page all_32576_to_00623.cpc0240.djvu
5.2 G-continued fractions......Page all_32576_to_00623.cpc0241.djvu
5.3 Generalized (or vector valued) continued fractions......Page all_32576_to_00623.cpc0244.djvu
Problems......Page all_32576_to_00623.cpc0246.djvu
Remarks......Page all_32576_to_00623.cpc0251.djvu
References......Page all_32576_to_00623.cpc0253.djvu
V Correspondence of continued fractions......Page all_32576_to_00623.cpc0257.djvu
1.1 Introducing the normed field......Page all_32576_to_00623.cpc0258.djvu
1.2 Correspondence at z = infty......Page all_32576_to_00623.cpc0259.djvu
1.3 Properties of the normed field (L, || \cdot ||)......Page all_32576_to_00623.cpc0262.djvu
2.1 Criteria for correspondence......Page all_32576_to_00623.cpc0264.djvu
2.2 Terminating continued fractions......Page all_32576_to_00623.cpc0267.djvu
2.4 C-fractions......Page all_32576_to_00623.cpc0268.djvu
2.5 When does f(z) have a regular C-fraction expansion?......Page all_32576_to_00623.cpc0272.djvu
2.6 Algorithms for producing corresponding continued fractions......Page all_32576_to_00623.cpc0275.djvu
3.1 Interpretation......Page all_32576_to_00623.cpc0281.djvu
3.2 A link between correspondence and classical convergence......Page all_32576_to_00623.cpc0286.djvu
4.1 A simple example......Page all_32576_to_00623.cpc0290.djvu
4.2 Approximants......Page all_32576_to_00623.cpc0293.djvu
4.3 Another example......Page all_32576_to_00623.cpc0295.djvu
Problems......Page all_32576_to_00623.cpc0297.djvu
Remarks......Page all_32576_to_00623.cpc0300.djvu
References......Page all_32576_to_00623.cpc0302.djvu
VI Hypergeometric functions......Page all_32576_to_00623.cpc0307.djvu
1.1 Why and how......Page all_32576_to_00623.cpc0308.djvu
1.2 A special case......Page all_32576_to_00623.cpc0312.djvu
1.3 Choice of approximants......Page all_32576_to_00623.cpc0314.djvu
1.4 Other continued fraction expansions......Page all_32576_to_00623.cpc0320.djvu
2.1 Notation......Page all_32576_to_00623.cpc0327.djvu
2.2 {}_1 F_1 (b;c;z)......Page all_32576_to_00623.cpc0328.djvu
2.3 {}_2 F_0 (a,b;z)......Page all_32576_to_00623.cpc0332.djvu
2.4 {}_0 F_1 (c;z)......Page all_32576_to_00623.cpc0333.djvu
3.1 Definition......Page all_32576_to_00623.cpc0334.djvu
3.2 {}_2 \phi_1 (a,b;c;q;z)......Page all_32576_to_00623.cpc0336.djvu
4.2 Some special cases......Page all_32576_to_00623.cpc0338.djvu
Problems......Page all_32576_to_00623.cpc0342.djvu
Remarks......Page all_32576_to_00623.cpc0344.djvu
References......Page all_32576_to_00623.cpc0345.djvu
VII Moments and orthogonality......Page all_32576_to_00623.cpc0347.djvu
1.1 Three examples......Page all_32576_to_00623.cpc0348.djvu
1.2 Moment sequences and moment functionals......Page all_32576_to_00623.cpc0354.djvu
1.3 Favard's theorem and Jacobi fractions......Page all_32576_to_00623.cpc0361.djvu
2.1 A quadrature formula......Page all_32576_to_00623.cpc0364.djvu
2.2 An example......Page all_32576_to_00623.cpc0367.djvu
3.1 The Stieltjes moment problem......Page all_32576_to_00623.cpc0369.djvu
3.2 Connection to continued fractions......Page all_32576_to_00623.cpc0372.djvu
Problems......Page all_32576_to_00623.cpc0377.djvu
Remarks......Page all_32576_to_00623.cpc0379.djvu
References......Page all_32576_to_00623.cpc0381.djvu
VIII Padé approximants......Page all_32576_to_00623.cpc0383.djvu
1.1 A creative problem......Page all_32576_to_00623.cpc0385.djvu
1.2 Padé approximants......Page all_32576_to_00623.cpc0390.djvu
1.3 Normal tables. Block structure......Page all_32576_to_00623.cpc0395.djvu
1.4 Connection to continued fraction expansions......Page all_32576_to_00623.cpc0398.djvu
1.5 A convergence result......Page all_32576_to_00623.cpc0401.djvu
2.1 Two-point Padé table......Page all_32576_to_00623.cpc0402.djvu
2.3 Multivariate Padé approximants......Page all_32576_to_00623.cpc0405.djvu
Problems......Page all_32576_to_00623.cpc0407.djvu
Remarks......Page all_32576_to_00623.cpc0408.djvu
References......Page all_32576_to_00623.cpc0409.djvu
IX Some applications in number theory......Page all_32576_to_00623.cpc0413.djvu
1.1 The Euclidean algorithm......Page all_32576_to_00623.cpc0415.djvu
1.2 Representation of positive numbers by regular continued fractions......Page all_32576_to_00623.cpc0418.djvu
1.3 Best approximation......Page all_32576_to_00623.cpc0424.djvu
2.1 Linear diophantine equations......Page all_32576_to_00623.cpc0426.djvu
2.2 Pell's equation......Page all_32576_to_00623.cpc0429.djvu
3.1 Introduction......Page all_32576_to_00623.cpc0434.djvu
3.2 Fermat factorization......Page all_32576_to_00623.cpc0436.djvu
3.3 Factor bases......Page all_32576_to_00623.cpc0439.djvu
3.4 A lemma on continued fractions......Page all_32576_to_00623.cpc0443.djvu
3.5 The continued fraction factoring algorithm......Page all_32576_to_00623.cpc0444.djvu
Problems......Page all_32576_to_00623.cpc0451.djvu
Remarks......Page all_32576_to_00623.cpc0453.djvu
References......Page all_32576_to_00623.cpc0455.djvu
X Zero-free regions......Page all_32576_to_00623.cpc0457.djvu
1.1 Introduction......Page all_32576_to_00623.cpc0458.djvu
1.2 An application of Van Vleck's theorem......Page all_32576_to_00623.cpc0464.djvu
1.3 An application of the parabola theorem......Page all_32576_to_00623.cpc0467.djvu
1.4 The Stieltjes case......Page all_32576_to_00623.cpc0469.djvu
1.5 The case when a_n \in R......Page all_32576_to_00623.cpc0472.djvu
1.6 A fundamental recurrence formula......Page all_32576_to_00623.cpc0476.djvu
1.7 Chain sequences......Page all_32576_to_00623.cpc0478.djvu
1.8 Two theorems on zero-free regions......Page all_32576_to_00623.cpc0480.djvu
2.1 Introductory remarks......Page all_32576_to_00623.cpc0484.djvu
2.2 Polynomials with real coefficients......Page all_32576_to_00623.cpc0486.djvu
2.3 Polynomials with complex coefficients......Page all_32576_to_00623.cpc0488.djvu
Problems......Page all_32576_to_00623.cpc0490.djvu
Remarks......Page all_32576_to_00623.cpc0493.djvu
References......Page all_32576_to_00623.cpc0495.djvu
XI Digital filters and continued fractions......Page all_32576_to_00623.cpc0497.djvu
1.1 Some introductory examples......Page all_32576_to_00623.cpc0498.djvu
1.2 Digital filters......Page all_32576_to_00623.cpc0500.djvu
1.3 Stable filters......Page all_32576_to_00623.cpc0505.djvu
1.4 Graph representation of filters......Page all_32576_to_00623.cpc0509.djvu
2.1 An old algorithm......Page all_32576_to_00623.cpc0517.djvu
2.2 Schur fractions and digital filters......Page all_32576_to_00623.cpc0521.djvu
3.1 General remarks......Page all_32576_to_00623.cpc0524.djvu
3.2 Stable filters with rational transfer function......Page all_32576_to_00623.cpc0525.djvu
Problems......Page all_32576_to_00623.cpc0530.djvu
Remarks......Page all_32576_to_00623.cpc0534.djvu
References......Page all_32576_to_00623.cpc0535.djvu
XII Applications to some differential equations......Page all_32576_to_00623.cpc0537.djvu
1.1 Introduction......Page all_32576_to_00623.cpc0539.djvu
1.2 An "almost" Euler-Cauchy equation......Page all_32576_to_00623.cpc0547.djvu
1.3 Two further examples......Page all_32576_to_00623.cpc0551.djvu
2.1 General Remarks......Page all_32576_to_00623.cpc0556.djvu
2.2 An old example......Page all_32576_to_00623.cpc0560.djvu
2.3 A new example......Page all_32576_to_00623.cpc0563.djvu
Problems......Page all_32576_to_00623.cpc0570.djvu
Remarks......Page all_32576_to_00623.cpc0572.djvu
References......Page all_32576_to_00623.cpc0573.djvu
Appendix. Some continued fraction expansions......Page all_32576_to_00623.cpc0575.djvu
1 Introduction......Page all_32576_to_00623.cpc0576.djvu
2 Mathematical constants......Page all_32576_to_00623.cpc0577.djvu
3.2 The exponential function......Page all_32576_to_00623.cpc0579.djvu
3.3 The general binomial function......Page all_32576_to_00623.cpc0580.djvu
3.4 The natural logarithm......Page all_32576_to_00623.cpc0582.djvu
3.5 Trigonometric and hyperbolic functions......Page all_32576_to_00623.cpc0584.djvu
3.6 Inverse trigonometric and hyperbolic functions......Page all_32576_to_00623.cpc0585.djvu
3.7 Continued fractions with simple values......Page all_32576_to_00623.cpc0587.djvu
4.1 General expressions......Page all_32576_to_00623.cpc0589.djvu
4.2 Special examples with {}_0 F_1......Page all_32576_to_00623.cpc0591.djvu
4.3 Special examples with {}_2 F_0......Page all_32576_to_00623.cpc0592.djvu
4.4 Special examples with {}_1 F_1......Page all_32576_to_00623.cpc0594.djvu
4.5 Special examples with {}_2 F_1......Page all_32576_to_00623.cpc0596.djvu
4.6 Some simple integrals......Page all_32576_to_00623.cpc0598.djvu
4.7 Gamma function expressions by Ramanujan......Page all_32576_to_00623.cpc0600.djvu
5.1 General expressions......Page all_32576_to_00623.cpc0609.djvu
5.2 Two general results by Andrews......Page all_32576_to_00623.cpc0610.djvu
5.3 q-expressions by Ramanujan......Page all_32576_to_00623.cpc0611.djvu
References......Page all_32576_to_00623.cpc0614.djvu
Subject index......Page all_32576_to_00623.cpc0617.djvu