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A First Course in Mathematical Analysis - J. C. Burkill

A First Course in Mathematical Analysis

Paperback Published: 12th February 1979
ISBN: 9780521294683
Number Of Pages: 196

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This straightforward course based on the idea of a limit is intended for students who have acquired a working knowledge of the calculus and are ready for a more systematic treatment which also brings in other limiting processes, such as the summation of infinite series and the expansion of trigonometric functions as power series. Particular attention is given to clarity of exposition and the logical development of the subject matter. A large number of examples is included, with hints for the solution of many of them.

'Books of this quality are rare enough to be hailed enthusiastically ... Essentially an introductory book for the mathematics specialist. But it is so fresh in conception and so lucid in style that it will appeal to anyone who has a general interest in mathematics.' The Times Educational Supplement 'This is an excellent book ... If I were teaching a course for honours students of the type described, this book would rank high as a possible choice of text.' Canadian Mathematical Bulletin ' ... it is a pleasure to be able to welcome a book on analysis written by an author who has a sense of style and who avoids the excessive use of symbolism which can make the subject unnecessarily difficult for the student.' Proceedings of the Edinburgh Mathematical Society

Prefacep. vii
Numbers
The branches of pure mathematicsp. 1
The scope of mathematical analysisp. 2
Numbersp. 3
Irrational numbersp. 7
Cuts of the rationalsp. 8
The field of real numbersp. 10
Bounded sets of numbersp. 13
The least upper bound (supremum)p. 15
Complex numbersp. 18
Modulus and phasep. 19
Sequences
Sequencesp. 23
Null sequencesp. 23
Sequence tending to a limitp. 25
Sequences tending to infinityp. 26
Sum and product of sequencesp. 28
Increasing sequencesp. 31
An important sequence a[superscript n]p. 32
Recurrence relationsp. 34
Infinite seriesp. 38
The geometric series [Sigma]x[superscript n]p. 39
The series [Sigma]n[superscript -k]p. 40
Properties of infinite scriesp. 43
Continuous Functions
Functionsp. 47
Behaviour of f(x) for large values of xp. 49
Sketching of curvesp. 49
Continuous functionsp. 51
Examples of continuous and discontinuous functionsp. 53
The intermediate-value propertyp. 56
Bounds of a continuous functionp. 57
Uniform continuityp. 60
Inverse functionsp. 62
The Differential Calculus
The derivativep. 65
Differentiation of sum, product, etc.p. 67
Differentiation of elementary functionsp. 69
Repeated differentiationp. 72
The sign of f'(x)p. 73
The mean value theoremp. 75
Maxima and minimap. 77
Approximation by polynomials. Taylor's theoremp. 78
Indeterminate formsp. 82
Infinite Series
Series of positive termsp. 88
Series of positive and negative termsp. 90
Conditional convergencep. 92
Series of complex termsp. 94
Power seriesp. 96
The circle of convergence of a power seriesp. 97
Multiplication of seriesp. 99
Taylor's seriesp. 101
The Special Functions of Analysis
The special functions of analysisp. 104
The exponential functionp. 105
Repeated limitsp. 105
Rate of increase of exp xp. 106
Exp x as a powerp. 107
The logarithmic functionp. 110
Trigonometric functionsp. 112
Exponential and trigonometric functionsp. 113
The inverse trigonometric functionsp. 116
The hyperbolic functions and their inversesp. 117
The Integral Calculus
Area and the integralp. 119
The upper and lower integralsp. 121
The integral as a limitp. 123
Continuous or monotonic functions are integrablep. 124
Properties of the integralp. 125
Integration as the inverse of differentiationp. 128
Integration by parts and by substitutionp. 129
The technique of integrationp. 131
The constant [pi]p. 136
Infinite integralsp. 137
Scries and integralsp. 140
Approximations to definite integralsp. 144
Approximations by subdivision. Simpson's rulep. 145
Functions of Several Variables
Functions of x and yp. 151
Limits and continuityp. 152
Partial differentiationp. 153
Differentiabilityp. 156
Composite functionsp. 158
Changes of variable. Homogeneous functionsp. 159
Taylor's theoremp. 163
Maxima and minimap. 164
Implicit functionsp. 166
Notes on the Exercisesp. 170
Indexp. 185
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521294683
ISBN-10: 0521294681
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 196
Published: 12th February 1979
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.1 x 14.38  x 1.32
Weight (kg): 0.27