Kennan T. Smith

Power Series from a Computational Point of View

• Broschiertes Buch

Produktbeschreibung

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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Produktdetails

• Universitext
• Verlag: Springer / Springer New York / Springer, Berlin
• Artikelnr. des Verlages: 978-0-387-96516-1
• 1987.
• Seitenzahl: 144
• Erscheinungstermin: 4. Mai 1987
• Englisch
• Abmessung: 235mm x 155mm x 9mm
• Gewicht: 229g
• ISBN-13: 9780387965161
• ISBN-10: 0387965165
• Artikelnr.: 24505567

Inhaltsangabe

1. Taylor Polynomials.- 1. Taylor Polynomials.- 2. Exponentials, Sines, and Cosines.- 3. The Geometric Sum.- 4. Combinations of Taylor Polynomials.- 5. Complex Taylor Polynomials.- Problems.- 2. Sequences and Series.- 1. Sequences of Real Numbers.- 2. Sequences of Complex Numbers and Vectors.- 3. Series of Real and Complex Numbers.- 4. Picard's Theorem on Differential Equations.- 5. Power Series.- 6. Analytic Functions.- 7. Preview.- Problems.- 3. Power Series and Complex Differentiability.- 1. Paths in the Complex Plane C.- 2. Path Integrals.- 3. Cauchy's Integral Theorem.- 4. Cauchy's Integral Formula and Inequalities.- Problems.- 4. Local Analytic Functions.- 1. Logarithms.- 2. Local Solutions to Analytic Equations.- 3. Analytic Linear Differential Equations.- Problems.- 5. Analytic Continuation.- 1. Analytic Continuation Along Paths.- 2. The Monodromy Theorem.- 3. Cauchy's Integral Formula and Theorem.- Problems.