1. |
Non‐linear Transformations of Divergent and Slowly Convergent Sequences |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 1-42
Daniel Shanks,
Preview
|
PDF (2173KB)
|
|
摘要:
This paper discusses a family of non‐linear sequence‐to‐sequence transformations designated asek, ekm, ẽk, anded. A brief history of the transforms is related and a simple motivation for the transforms is given. Examples are given of the application of these transformations to divergent and slowly convergent sequences. In particular the examples include numerical series, the power series of rational and meromorphic functions, and a wide variety of sequences drawn from continued fractions, integral equations, geometry, fluid mechanics, and number theory. Theorems are proven which show the effectiveness of the transformations both in accelerating the convergence of (some) slowly convergent sequences and in inducing convergence in (some) divergent sequences. The essential unity of these two motives is stressed. Theorems are proven which show that these transforms often duplicate the results of well‐known, but specialized techniques. These special algorithms include Newton's iterative process, Gauss's numerical integration, an identity of Euler, the Padé Table, and Thiele's reciprocal differences. Difficulties which sometimes arise in the use of these transforms such as irregularity, non‐uniform convergence to the wrong answer, and the ambiguity of multivalued functions are investigated. The concepts of antilimit and of the spectra of sequences are introduced and discussed. The contrast between discrete and continuous spectra and the consequent contrasting response of the corresponding sequences to thee1transformation is indicated. The characteristic behaviour of a semiconvergent (asymptotic) sequence is elucidated by an analysis of its spectrum into convergent components of large amplitude and divergent components of sma
ISSN:0097-1421
DOI:10.1002/sapm19553411
年代:1955
数据来源: WILEY
|
2. |
Bounds of the Legendre Function |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 43-49
Keeve M. Siegel,
Preview
|
PDF (351KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534143
年代:1955
数据来源: WILEY
|
3. |
Two Formulae Involving Generalized Legendre Functions of Integral and Nonintegral Index |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 50-53
George C. Sponsler,
Preview
|
PDF (224KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534150
年代:1955
数据来源: WILEY
|
4. |
Equally Weighted Quadrature Formulas over Semi‐infinite and Infinite Intervals |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 54-63
Herbert E. Salzer,
Preview
|
PDF (580KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534154
年代:1955
数据来源: WILEY
|
5. |
TheN‐Step Iteration Procedures |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 64-73
Edward J. Craig,
Preview
|
PDF (497KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534164
年代:1955
数据来源: WILEY
|
6. |
Singular Surfaces in Plasticity |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 74-79
J. L. Ericksen,
Preview
|
PDF (370KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534174
年代:1955
数据来源: WILEY
|
7. |
Topological and Dynamical Invariant Theory of an Electrical Network |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 81-94
Thomas C. Doyle,
Preview
|
PDF (887KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534181
年代:1955
数据来源: WILEY
|
8. |
Distribution of Sizes of Spheres in a Solid from a Study of Slices of the Solid |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 95-102
Walter P. Reid,
Preview
|
PDF (413KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm195534195
年代:1955
数据来源: WILEY
|
9. |
Osculatory Quadrature Formulas |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 103-112
Herbert E. Salzer,
Preview
|
PDF (495KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm1955341103
年代:1955
数据来源: WILEY
|
10. |
A Family of Plane Compressible Flows Past a Certain Semi‐Infinite Body |
|
Journal of Mathematics and Physics,
Volume 34,
Issue 1,
1955,
Page 113-118
A. R. Manwell,
Preview
|
PDF (324KB)
|
|
ISSN:0097-1421
DOI:10.1002/sapm1955341113
年代:1955
数据来源: WILEY
|