MA258 Mathematical Analysis Iii Notes | Assignment Help | Syllabus
Country: United Kingdom
University: The University Of Warwick
The following three concepts are covered: (1) integration, (2) convergence of sequences and series of functions, and (3) norms.
The goal of integration is to find the region under a curve. The basic theorem of calculus explains how integration and differentiation are related. Integration, on the other hand, necessitates the application of a limit, and the deeper characteristics of integration need a precise and thorough examination of this limiting process. This lesson establishes that every continuous function may be integrated, as well as the calculus basic theorem. It also goes through how integration may be used to define and establish the fundamental characteristics of some of the most basic functions of analysis.
Many functions may be expressed as sums of series or as limits of sequences of simpler functions: for example, a power series is a limit of polynomials, and a Fourier series is the sum of a trigonometric series with coefficients determined by specific integrals. The module’s second section provides techniques for determining if a function defined as the limit of a series of other functions is continuous, differentiable, and integrable, as well as methods for differentiating and integrating this limit. Norms are employed at several stages before being used to prove that a Differential Equation has a solution. Aims:
To get a solid understanding of how the integral of regulated functions is constructed;
to investigate the limit of a uniformly convergent series of functions’ continuity, differentiability, and integral;
to explain convergence and continuity in a vector space using the notion of norm
Recognize the need of a rigorous theory of integration for regulated functions, and that this may be constructed by approximating the area under the graph with rectangles;
Discusses uniform and pointwise convergence of functions, as well as limit function characteristics;
be able to demonstrate the major integration outcomes: The Fundamental Theorem of Calculus states that every continuous function may be integrated on a limited interval.
The Contraction Mapping Theorem is proved and applied.
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