Proseminar und Seminar : Asymptotische Analysis 2009/2010
Proseminar
Presentation : Asymptotic analysis is mainly concerned with real-valued functions of a real variable. The idea is to study the limit behaviour of such functions as the real variable x tends to +∞. More precisely, in the case where the functions diverge -i.e. tends to infinity as x does- we want to compare their rate of growth. For example, the identity function f(x) = x tends to ∞ more slowly than the square function g(x) = x2, the two of them being polynomial functions. But the exponential function grows faster than any polynomial function, any of which, in turn, grows faster than the logarithmic function.
During the fourteen sessions of this proseminar, we will introduce the notions of comparison relations and asymptotic scales. In particular, we will introduce and investigate the fundamental scale of logarithmic-exponential functions of G.H. Hardy. We will also investigate general properties of these objects under the fundamental operations, namely addition, multiplication, derivation and integration. Thus we will study applications to the study of convergence of sequences and series, and to the approximation of solutions to differential equations.
1. What is a comparison relation ? Part 1, by Jessica Müller (pdf)
2. What is a comparison relation ? Part 2, by Carola Hehmann (pdf)
3. What is an asymptotic scale ?, by Karin Ott (pdf)
4. Theorems of du Bois-Reymond, by Simone Metzdorf (pdf)
5. and 6. The logarithmic-exponential (LE) functions - Part 1 and 2, by Jonjer Jennerjahn and Nicola Kreutzer (pdf)
Seminar
Presentation : In this seminar, we will resume the "Introduction to Asymptotic Analysis" started in the winter semester proseminar. The purpose is as before to go through the basic material for Asymptotic Analysis, following some chapters of the reference book of G.H. Hardy: "Orders of Infinity; the inifinitärcalcül of Paul tu Bois-Reymond" (Cambridge University Press, 1954).
During the winter semester proseminar, we introduced the following basic ideas. Asymptotic analysis is mainly concerned with real-valued functions of a real variable. We studied the limit behaviour of such functions as the real variable
x tends to + ∞. More precisely, in the case where the functions diverge -i.e. tends to infinity as x does- we compared their rate of growth. For example, the identity function f (x) = x tends to ∞ more slowly than the square function g(x) = x2 ,the two of them being polynomial functions. But the exponential function grows faster than any polynomial function, any of which, in turn, grows faster than the logarithmic function.
We introduced the notions of comparison relations and asymptotic scales. In particular, we introduced and investigated the fundamental scale of logarithmic-exponential functions (L-functions) of G.H. Hardy. During this summer semester seminar, in the context of these L-functions, we will introduce the notion of comparability classes. We will study also functions which do not conform to any L-function. Moreover, we will be interested in understanding the behavior of L-functions under the fundamental operations of differentiation and integration. This will lead finally us to various important applications about convergence of series, asymptotic expansions, analytic functions and differential equations.
1. What is a comparison class? Nicolas Kreutzer - Jonjer Jennerjahn. 21 April 2010 (pdf)
2. Functions which do not conform to log-exp functions: oscillating functions. Jonjer Jennerjahn - Nicolas Kreutzer. 28 April 2010 (pdf)
3. Functions which do not conform to log-exp function: transexponentials and cislogarithms. Simone Metzdorf. 05 May 2010 (pdf)
4. Integration and derivation. Jessica Müller. 12 May 2010 (pdf)
5. Application 1: Riemann’s convergence criterion for series. Karin Ott 26 May 2010 (pdf)
6. Application 2: Stirling’s formula. Carola Hehmann. 02 June 2010 (pdf)
7. Application 3: Power series, entire functions. Maria Havers. 09 June 2010 (pdf)
8. Application 4: approximation of solutions to differential equations. Charu Goel. 16 June 2010 (pdf)
Main reference book:
G.H. Hardy, Orders of Infinity - The "Infinitärcalcül" of Paul du Bois-Reymond.
Cambridge at the University Press, 1954.
Complementary books:
J. Dieudonné , Calcul infinitésimal, (1968), ed. Hermann
H.J. Keisler, Elementary calculus. available freely on internet.
M. Barner, F. Florh, Analysis 1. ed. De Gruyter Lehrbuch, 1974.
C. Blatter, Analysis 1. ed. Heidelberger Taschenbcher 151, Springer-Verlag 1974.
Any other good book of Elementary Calculus and/or Analysis 1...