ExAn1 - Analysis I Prof. T. W. K¨ orner Lent 2003 Contents...

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Unformatted text preview: Analysis I Prof. T. W. K¨ orner Lent 2003 Contents 1 Why do we bother? 2 2 The axiom of Archimedes 3 3 Series and sums 6 4 Least upper bounds 10 5 Continuity 14 6 Differentiation 18 7 The mean value theorem 22 8 Complex variable 27 9 Power series 29 10 The standard functions 32 11 Onwards to the complex plane 40 12 The Riemann integral 45 13 Some properties of the integral 52 14 Infinite integrals 59 15 Further reading 63 1 Lecturer’s preamble This course centres around 1 idea, the re-founding of calculus on a rigorous basis. This is essentially based on the definition of a limit: ∀ > ∃ n ( ) such that | a n- a | < ∀ n ≥ n ( ) From this we get the FUNDAMENTAL THEOREM OF ANALYSIS Every strictly increasing sequence bounded above tends to a limit This does not work in Q , e.g. look at the decimal expansion of √ 2, as you add more and more digits it is clearly increasing, it is bounded above by 2, but its limit ( √ 2) does not exist in Q . However, it always works on R Using this axiom and the laws of algebra (e.g. a + b = b + a ), we can now refound the calculus. NOTE: We can now split proofs and results into two groups, those which are mere algebra (i.e. proof also works on Q ) and those using analysis. 1 Why do we bother? It is surprising how many people think that analysis consists in the difficult proofs of obvious theorems. All we need know, they say, is what a limit is, the definition of continuity and the definition of the derivative. All the rest is ‘intuitively clear’. If pressed they will agree that these definitions apply as much to the rationals Q as to the real numbers R . They then have to explain the following interesting example. Example 1.1. If f : Q → Q is given by f ( x ) =- 1 if x 2 < 2, f ( x ) = 1 otherwise, then (i) f is a continuous function with f (0) =- 1, f (2) = 1 yet there does not exist a c with f ( c ) = 0, (ii) f is a differentiable function with f ( x ) = 0 for all x yet f is not constant. What is the difference between R and Q which makes calculus work on one even though it fails on the other? Both are ‘ordered fields’, that is, both support operations of ‘addition’ and ‘multiplication’ together with a relation ‘greater than’ (‘order’) with the properties that we expect. If the reader is interested she will find a complete list of the appropriate axioms in texts like the altogether excellent book of Spivak [ 5 ] and its many rather less excellent competitors, but, interesting as such things may be, they are irrelevant to our purpose which is not to consider the shared properties of R and Q but to identify a difference 2 between the two systems which will enable us to exclude the possibility of a function like that of Example 1.1 for functions from R to R ....
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This note was uploaded on 02/18/2012 for the course MATH 533 taught by Professor Drewarmstrong during the Fall '11 term at University of Miami.

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ExAn1 - Analysis I Prof. T. W. K¨ orner Lent 2003 Contents...

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