fractional linear transformation

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1: 19.14 Reduction of General Elliptic Integrals

… ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …

2: 1.9 Calculus of a Complex Variable

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Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …

3: 15.8 Transformations of Variable

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. …

4: 15.19 Methods of Computation

… ►For

z\in\mathbb{R}

it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval

[0,\frac{1}{2}]

. ►For

z\in\mathbb{C}

it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when

z={\mathrm{e}}^{\pm\pi\mathrm{i}/3}

. … ►When

\Re z>\frac{1}{2}

it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8). … ►

§15.19(v) Continued Fractions

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5: Bibliography S

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D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:: MathReview (J. Kuntzmann), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

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G. E. Shilov (2013) Introduction to the Theory of Linear Spaces. Martino, Mansfield Center, CT.

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Notes:: republication of 1961 first edition, Prentice-Hall,New Jersey
External Links:: ISBN 978-1-61427-457-5, MathReview Entry
Referenced by:: §1.18(i), §1.2(vi), §1.3(iv).
Encodings:: BibTeX

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B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:: Document, MathReview (J. D. DePree), ZentralBlatt
Referenced by:: §31.10(ii).
Encodings:: BibTeX

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R. Spigler and M. Vianello (1992) Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.

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External Links:: ISSN 0377-0427, Document, MathReview (Wan Biao Ma), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

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M. H. Stone (1990) Linear transformations in Hilbert space. American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.

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Notes:: Reprint of the 1932 original
External Links:: ISBN 0-8218-1015-4, Document, Link, MathReview (P. R. Halmos)
Referenced by:: §1.18(x).
Encodings:: BibTeX

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6: Bibliography G

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L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:: FullText, MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §35.2, §35.3(ii).
Encodings:: BibTeX

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W. Gautschi (1997b) The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:: ISBN 90-5699-521-9, MathReview, ZentralBlatt
Referenced by:: §3.6(iii), §3.6(vii).
Encodings:: BibTeX

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A. Gil and J. Segura (2003) Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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External Links:: ISSN 0036-1429, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi), §3.8(v).
Encodings:: BibTeX

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J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:: ISBN 0-8176-3837-7, MathReview, ZentralBlatt
Referenced by:: §15.17(v).
Encodings:: BibTeX

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E. P. Gross and S. Ziering (1958) Kinetic theory of linear shear flow. Phys. Fluids 1 (3), pp. 215–224.

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Referenced by:: §18.39(iii).
Encodings:: BibTeX

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7: Errata

… ►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. … ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

… ►

Subsection 13.29(v)

A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

… ►

Subsection 15.19(v)

A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

…

8: Bibliography W

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Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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External Links:: ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

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Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:: ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

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W. Wasow (1985) Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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External Links:: ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: Ch.9.
Encodings:: BibTeX

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G. B. Whitham (1974) Linear and Nonlinear Waves. John Wiley & Sons, New York.

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Notes:: Reprinted in 1999.
External Links:: ISBN 0471359424, MathReview, ZentralBlatt
Referenced by:: §9.16.
Encodings:: BibTeX

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R. Wong (2014) Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12 (4), pp. 463–484.

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External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §2.9(iii), §2.9(iii).
Encodings:: BibTeX

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9: Bibliography O

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A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:: ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).
Encodings:: BibTeX

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A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

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A. B. Olde Daalhuis and F. W. J. Olver (1998) On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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External Links:: ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt
Referenced by:: §16.25, §2.7(ii), §3.7(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1995) Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:: ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §10.17(v), §10.40(iv), §2.11(v), §9.7(v).
Encodings:: BibTeX

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F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:: FullText, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

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10: Bibliography M

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J. P. McClure and R. Wong (1979) Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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External Links:: ISSN 0020-2932, Document, MathReview, ZentralBlatt
Referenced by:: §2.6(iii).
Encodings:: BibTeX

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J. C. P. Miller (1950) On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.

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External Links:: Document, MathReview (W. R. Wasow), ZentralBlatt
Referenced by:: §12.2(vi), §2.7(iv), §2.7(iv).
Encodings:: BibTeX

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K. S. Miller and B. Ross (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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External Links:: ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).
Encodings:: BibTeX

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C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

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External Links:: ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

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C. Mortici (2013a) A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt
Referenced by:: §5.10.
Encodings:: BibTeX

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