Taylor Series, Partial Fractions, Laurent Series, and Residues - Couverture souple

Laurent, Hermann; Staines, Jeremy

Couverture souple

ISBN 13 : 9798656821261

Editeur : Independently published, 2020

Synopsis

Hermann Laurent (1853), who is not the Laurent associated with the Laurent Series (Pierre Alphonse Laurent), gives us a formal method for the decomposition of rational functions into partial fractions. He deals with simple and multiple roots, both real and imaginary. His method depends upon the Taylor Series, which is hence the first topic to be discussed. He also gives an error analysis of the Taylor series. Partial fractions find use in the ordinary integral calculus. From here I introduce the Laurent Series expression of a rational function, according to a selected root of its denominator. To achieve this, the remainder of the function must be expressed in terms of this selected root; this too is dependent upon a Taylor series. From the Laurent series I show the importance of the residue, for the integration of rational functions around a closed contour which circumscribes the selected root, in the complex plane. Two applications of this, for which I give simple examples, are: the inversion of Laplace Transforms, and the evaluation of real definite integrals. And, given that all the above depends upon knowledge of roots of equations, I have included formulae, provided by Camillo Pagliani (1849), for the roots of quadratic, cubic, and quartic polynomials. You will require a good knowledge of calculus and complex variables to understand this work. Some familiarity with Laplace Transforms would be of advantage, though perhaps not strictly necessary.

Note: For an alternative proof of an integral found on p23 of this booklet, see home.lizzy.com.au/jeremy.staines/Stress_Integral.pdf

Détails de l'édition

• Éditeur: Independently published
• Date d'édition: 2020
• Langue: anglais
• ISBN 13: 9798656821261
• Reliure: Broché
• Nombre de pages: 44
• Coordonnées du fabricant: Printed by Amazon on behalf of the author; https://www.amazon.fr/hz/contact-us; ; c/o Amazon Media EU S.à.r.l., 38 Avenue John F. Kennedy; Luxembourg; L-1855; Luxembourg