Power Series from a Computational Point of View:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1987
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | At the end of the typical one quarter course on power series the students lack the means to decide 2 whether 1/(1+x ) has an expansion around any point ~ 0, or the tangent has an expansion anywhere and the means to evaluate and predict errors. In using power series for computation the main problems are: 1) To predict a priori the number N of terms needed to do the computation with a specified accuracy; and 2) To find the coefficients aO, •.• ,a • N These are the problems addressed in the book. Typical computations envisioned are: -6 calculate with error ~ 10 the integrals If/2 J (If/2-x)tan x dx o or the solution to the differential equation 2 y"+(sin x)Y'+x y = 0, y(O) = 0, y'(O) 1, on the interval 0 ~ x ~ 1. This computational point of view may seem narrow, but, in fact, such computations require the understa- ing and use of many of the important theorems of ele mentary analytic function theory: Cauchy's Integral Theorem, Cauchy's Inequalities, Unique Continuation, Analytic Continuation and the Monodromy Theorem, etc. The computations provide an effective motivation for learning the theorems and a sound basis for understa- ing them. To other scientists the rationale for the vi computational point of view might be the need for ef- cient accurate calculation; to mathematicians it is the motivation for learning theorems and the practice with inequalities, ~'s, o's, and N's |
| Physical Description: | 1 Online-Ressource (VIII, 132 p) |
| ISBN: | 9781461395812 9780387965161 |
| ISSN: | 0172-5939 |
| DOI: | 10.1007/978-1-4613-9581-2 |
Staff View
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| 500 | |a At the end of the typical one quarter course on power series the students lack the means to decide 2 whether 1/(1+x ) has an expansion around any point ~ 0, or the tangent has an expansion anywhere and the means to evaluate and predict errors. In using power series for computation the main problems are: 1) To predict a priori the number N of terms needed to do the computation with a specified accuracy; and 2) To find the coefficients aO, •.• ,a • N These are the problems addressed in the book. Typical computations envisioned are: -6 calculate with error ~ 10 the integrals If/2 J (If/2-x)tan x dx o or the solution to the differential equation 2 y"+(sin x)Y'+x y = 0, y(O) = 0, y'(O) 1, on the interval 0 ~ x ~ 1. This computational point of view may seem narrow, but, in fact, such computations require the understa- ing and use of many of the important theorems of ele mentary analytic function theory: Cauchy's Integral Theorem, Cauchy's Inequalities, Unique Continuation, Analytic Continuation and the Monodromy Theorem, etc. The computations provide an effective motivation for learning the theorems and a sound basis for understa- ing them. To other scientists the rationale for the vi computational point of view might be the need for ef- cient accurate calculation; to mathematicians it is the motivation for learning theorems and the practice with inequalities, ~'s, o's, and N's | ||
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Record in the Search Index
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:07Z |
| institution | BVB |
| isbn | 9781461395812 9780387965161 |
| issn | 0172-5939 |
| language | English |
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| spelling | Smith, Kennan T. Verfasser aut Power Series from a Computational Point of View by Kennan T. Smith New York, NY Springer New York 1987 1 Online-Ressource (VIII, 132 p) txt rdacontent c rdamedia cr rdacarrier Universitext 0172-5939 At the end of the typical one quarter course on power series the students lack the means to decide 2 whether 1/(1+x ) has an expansion around any point ~ 0, or the tangent has an expansion anywhere and the means to evaluate and predict errors. In using power series for computation the main problems are: 1) To predict a priori the number N of terms needed to do the computation with a specified accuracy; and 2) To find the coefficients aO, •.• ,a • N These are the problems addressed in the book. Typical computations envisioned are: -6 calculate with error ~ 10 the integrals If/2 J (If/2-x)tan x dx o or the solution to the differential equation 2 y"+(sin x)Y'+x y = 0, y(O) = 0, y'(O) 1, on the interval 0 ~ x ~ 1. This computational point of view may seem narrow, but, in fact, such computations require the understa- ing and use of many of the important theorems of ele mentary analytic function theory: Cauchy's Integral Theorem, Cauchy's Inequalities, Unique Continuation, Analytic Continuation and the Monodromy Theorem, etc. The computations provide an effective motivation for learning the theorems and a sound basis for understa- ing them. To other scientists the rationale for the vi computational point of view might be the need for ef- cient accurate calculation; to mathematicians it is the motivation for learning theorems and the practice with inequalities, ~'s, o's, and N's Mathematics Global analysis (Mathematics) Analysis Mathematik Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Potenzreihenentwicklung (DE-588)4175496-7 gnd rswk-swf Analytische Funktion (DE-588)4142348-3 gnd rswk-swf Potenzreihe (DE-588)4138577-9 gnd rswk-swf Polynom (DE-588)4046711-9 gnd rswk-swf Potenzreihe (DE-588)4138577-9 s Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 Potenzreihenentwicklung (DE-588)4175496-7 s 2\p DE-604 Polynom (DE-588)4046711-9 s 3\p DE-604 Analytische Funktion (DE-588)4142348-3 s 4\p DE-604 https://doi.org/10.1007/978-1-4613-9581-2 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Smith, Kennan T. Power Series from a Computational Point of View Mathematics Global analysis (Mathematics) Analysis Mathematik Numerisches Verfahren (DE-588)4128130-5 gnd Potenzreihenentwicklung (DE-588)4175496-7 gnd Analytische Funktion (DE-588)4142348-3 gnd Potenzreihe (DE-588)4138577-9 gnd Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4175496-7 (DE-588)4142348-3 (DE-588)4138577-9 (DE-588)4046711-9 |
| title | Power Series from a Computational Point of View |
| title_auth | Power Series from a Computational Point of View |
| title_exact_search | Power Series from a Computational Point of View |
| title_full | Power Series from a Computational Point of View by Kennan T. Smith |
| title_fullStr | Power Series from a Computational Point of View by Kennan T. Smith |
| title_full_unstemmed | Power Series from a Computational Point of View by Kennan T. Smith |
| title_short | Power Series from a Computational Point of View |
| title_sort | power series from a computational point of view |
| topic | Mathematics Global analysis (Mathematics) Analysis Mathematik Numerisches Verfahren (DE-588)4128130-5 gnd Potenzreihenentwicklung (DE-588)4175496-7 gnd Analytische Funktion (DE-588)4142348-3 gnd Potenzreihe (DE-588)4138577-9 gnd Polynom (DE-588)4046711-9 gnd |
| topic_facet | Mathematics Global analysis (Mathematics) Analysis Mathematik Numerisches Verfahren Potenzreihenentwicklung Analytische Funktion Potenzreihe Polynom |
| url | https://doi.org/10.1007/978-1-4613-9581-2 |
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