Looking for a Time Series Analysis Text

In summary, "Analysis by Its History" by E. Hairer and G. Wanner is a valuable resource for those studying undergraduate level Analysis. It places analysis within its historical context and covers topics chronologically, providing a different approach to traditional textbooks. While it may not be suitable as a main textbook, it serves as a helpful supplement for better understanding the subject. Another recommended book for analysis is "A First Course in Mathematical Analysis" by David Alexander Brannan. It includes worked out problems and solutions, making it a great resource for self-study.

For those who have used this book

  • Lightly don't Recommend

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  • Strongly don't Recommend

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  • Total voters
    3
  • #1
Dj Sneaky Whiskers
Analysis by Its History

E. Hairer & G. Wanner

Published by Springer, as part of their Undergraduate Texts in Mathematics series.

ISBN 0-387-94551-2

http://www.springer-ny.com


Introduction: For many first year Mathematics undergraduates an introduction to Analysis is often accompanied by a sensation verging close to shell shock. One moment they are at one with the world, enjoying watching their familiarity with the Calculus and Linear Algebra, first established at high school, bloom and grow. The next moment they find themselves trapped on foreign soil, menaced by predicates, quantifiers and a professor hurling propositions and connectives at them with seemingly endless ferocity.

This may be somewhat of an exaggeration, but nevertheless it is undoubtably the case that one of the major hurdles that faces the student in understanding and gaining confidence with Analysis is the apparent 'disconnected' and almost artificial nature of the subject. This is where 'Analysis by Its History' by E. Hairer and G. Wanner comes into lend a helping hand.

Audience: Mainly of interest to those studying undergraduate level Analysis, although possibly also having some appeal for those studying the natural sciences and/or engineering subjects who are curious about the development and definitions of the mathematics they regularly use.

Pros: Places analysis within its historical context, illustrating the connections between definitions and concepts used in analysis with the practical problems that led to their formulation. Covers topics in Analysis chronologically (i.e. beginning with the infinite series as they first arose in antiquite, progressing to the established mathematical rigour of the 19th century), thus offering a different method of presentation to the conventional 'back-to-front' method.
Clearly written with many illustrative examples and a vast number of historical quotes begging to be appropriated for use on personal websites.

Cons: Occassionally topics are covered slightly too quickly, with scant mention to or explanation of set theory and logic. Cursed with the unspeakably vile yellow cover of Springer publications (and don't even begin to pretend that one of the concerns when buying textbooks isn't how cool it will look on your bookshelf).

Conclusion: A valuable addition to the library of anyone wanting to study pure mathematics but is left feeling uncertain and lacking confidence in their understanding of analysis by lectures and other textbooks. Having said this, however, 'Analysis by Its History' cannot be recommended as a 'main' textbook for the subject. It proves most useful in setting topics covered in more orthodox books, such as 'A First Course in Mathematical Analysis' by J.C. Burkill, or even online lecture notes (I recommend those of Vitali Liskevich of the University of Bristol for a good coverage of set theory, logic, and analysis. But, you'll understand, I am exceptionally biased in this matter), in some kind of wider context.
 
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  • #2
"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher order mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced." --Introduction to Mathematical philosophy, Russell.

I'm currently on the more familiar path, but the books my school uses are mostly gearded towards problem solving with very little theory in calculus I - III. My school library has books by Apostol and Spivak, but both of them look like difficult books to try and start learning analysis. Aren't there books that can help ease my way into the subject of higher-order mathematics?

Also, I've been wanting to study more set theory and logic theory. I've been trying to read some Bertrand Russell books on logic and mathematics, like "On Denoting" and "The Logic of Relations," but I have no idea about Peano notation and thus cannot even get past the third page in the latter essay. Even his Introduction to Mathematical Philosophy is a bit confusing, I'm simply not too familiar with using methods that look like something from the binomial theorem to prove things, especially in regards to classes. Russell would define things like "1" as the class of all unit classes.

I have given up on these more complicated Russell books but would still like to learn some fundamentals of arithmetic and peano notation.

Can you get off the ground in set theory without worrying about the philosophical mathematics stuff?

Also, what is a good generalized overview of mathematics:

Principles of Mathematics (russell) or An Introduction to Mathematics (whitehead)
 
  • #3
I'm thinking about getting this book. I'm a physics major, and I think the only analysis course I'm required to take later as a prerequisite for graduate courses is Introduction to Complex Analysis. So far, I've taken Cal I-III and Linear Algebra. Differential Equations will probably be in the fall. Do I have enough knowledge so far to try to tackle this book for fun? Because I'm required to take so many courses for my physics major, I only need one more to get a math minor, which is what I'm doing.

https://www.amazon.com/dp/0486406830/?tag=pfamazon01-20
 
  • #4
For anyone out there look for a good book on mathematical analysis the following is really excellent.

A First Course in Mathematical Analysis by David Alexander Brannan

This is based on a course given by the open university in the UK and it really is a wonderful introduction to analysis. All problems are worked out and solutions to all problems are given. This in addition to the classic Spivak is perhaps the best introduction your going to get on mathematical analysis. This book is nicely broken down into lots of small sections with examples, problems and solutions. Carefully working through this book will pay off on the long term.

Cheers
David.
 
  • #5
Hey, I am taking a course in Vector analysis and field theory and I was wondering if anycone can recommend a text/work-book.


Course content:


Scalar and vector fields. Scaling and dimensionless parameters. Equiscalar lines and surfaces, streamlines, field lines. Gradient vector, curl and divergence of vectors in Cartesian and polar coordinates with physical interpretation. Vector flux and circulation. Curve integrals, surface integrals, computation of pressure force. The divergence theorem (Gauss' theorem), Stokes' and Green's theorems. Mass conservation and equations of motion for fluids. Potential flows (source, sink, point vortex and dipole). Bernoulli's equation for stationary inviscid fluid flow. Convective and conductive heat flux, Fourier's law, equations for heat transport in fluids and solids. Use of Matlab for visualization of fields.

Learning outcomes
To give an introduction in field theory (vector calculus) with applications related mainly to fluid mechanics, geophysics, and physics. Through exercises using both analytical, numerical methods and computer graphics the students shall become familiar with the fundamental methods in field theory and its applications. The subject gives the students a good basis for further studies in mechanics and applied mathematics, physics, geosciences and astrophysics.
 
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  • #6
I need to learn principal component analysis from scratch for my research project, but it seems like a fairly staple technique that there isn't a text on it which stands out in particular... I found one by Jolliffe and another by Izenman (Modern Multivariate Statistical Techniques, Springer) with a chapter on linear dimensional reduction and principal component analysis.

Is there anything you would recommend me? Also bearing in mind, I don't have much background in applied statistics...

Thanks in advance!
 
  • #7
In the second edition of Apostol's Mathematical Analysis he comments:

"The material on line integrals, vector analysis, and surface integrals has been deleted."

The contents from the 1st Edition (I think, stolen from an Amazon comment) are:

Real and Complex Number Systems;
Fundamental Notions of Set Theory;
Elements of Point-Set Theory;
The Concepts of Limit and Continuity;
Differentiation of Real-Valued Functions;
Differentiation of Functions of Several Variables;
Applications of Partial Differentiation;
Functions of Bounded Variation, Rectifiable Curves, and Connected Sets;
Riemann-Stieltjes Integration Theory;
Multiple Integrals and Line Integrals;
Vector Analysis;
Infinite Series and Products;
Sequences of Functions;
Improper Riemann-Stieltjes Integrals;
Fourier Series and Fourier Integrals;
Cauchy's Theorem and Calculus of Residues;

& the contents of the second edition are:

Real and Complex Number Systems;
Fundamental Notions of Set Theory;
Elements of Point-Set Theory;
Limits and Continuity;
Derivatives;
Functions of Bounded Variation & Rectifiable Curves;
The Riemann-Stieltjes Integral;
Infinite Series and Products;
Sequences of Functions;
The Lebesgue Integral;
Fourier Series and Fourier Integrals;
Multivariable Differential Calculus
Implicit Functions & Extremum Problems
Multiple Riemann Integrals;
Multiple Lebesgue Integrals;
Vector Analysis;
Cauchy's Theorem and the Residue Calculus;

so really the only major differences are the exclusion of vector analysis, line & surface integrals to what you gain in Lebesgue integration.

My question is whether the material on line & surface integrals & the vector analysis material from the first edition is any better than the exposition he gives of in his Calculus II?

Keep in mind that the differences between the first edition of his calculus & second aren't really that major, a bit more linear algebra (I only browsed the book but it looked pretty close to the second edition) but the difference between his Calculus & Mathematical Analysis books is pretty big so I really have no idea whether he approaches the material from a higher perspective. If you've ever compared the old & new editions of volume 2 of Courant you'll see the drastic change in the approach to the material so I can only hope for a similar occurrence with Apostol.
 
  • #8
Hi!
I am currently planning to get a book on Real Analysis for self
studying before diving into my 4th year real analysis course.
The standard textbook for my 4th year course is Stein's Measure,
but I do not like much about abstract measure introduced near the end.
Perhaps because I am currently taking 3rd year real analysis course
in the level of Pugh with some other additional materials.

Anyway, I am considering one of the followings:
Folland - Real Analysis
Bruckner, Bruckner, Thomson - Real Analysis
Yeh - Real Analysis
Kantorovitz - Introduction to Modern Analysis
(and maybe Cohn - Measure Theory)

(Note: Royden is omitted because I am waiting for 2nd printing
and waiting so that I can get it cheap from some website
(like abebooks), so 12 pages of erratas are all fixed)

Which book do you think is most suitable for self-study?
 
  • #9

Table of Contents:
Code:
[LIST]
[*] The Real and Complex Number Systems
[LIST]
[*] Introduction
[*] The field axioms
[*] The order axioms
[*] Geometric representation of real numbers
[*] Intervals
[*] Integers
[*] Tim unique factorization theorem for integers
[*] Rational numbers
[*] Irrational numbers
[*] Upper bounds, maximum element, least upper bound (supremum) 
[*] The completeness
[*] Some properties of the supremum
[*] Properties of the integer deduced from the completeness axiom
[*] The Archimedean property of the real-number system
[*] Rational numbers with finite decimal representation
[*] Finite decimal approximations to real mumbers
[*] Infinite decimal representation of real numbers
[*] Absolute values and the triangle inequality
[*] The Cauchy-Schwarz inequality
[*] Plus and minus infinity and the extended real number system R*
[*] Complex numbers
[*] Geometric representation of complex numbers
[*] The imaginary unit
[*] Absolute value of a complex number
[*] Impossibility of ordering the complex numbers
[*] Complex exponentials
[*] Further properties of complex exponentials
[*] The argument of a complex number
[*] Integral powers and roots of complex numbers
[*] Complex logarithms
[*] Complex powers
[*] Complex sines and cosines
[*] Infinity and the extended complex plane C*
[*] Exercises
[/LIST]
[*] Some Basic Notions of Set Theory
[LIST]
[*] Introduction
[*] Notations
[*] Ordered pairs
[*] Certesian product of two sets
[*] Relations and functions
[*] Further terminology concerning functions
[*] One-to-one functions and inverses
[*] Composite functions
[*] Sequences
[*] Similar (equinumerous) sets
[*] Finite and infinite sets
[*] Countable and uncountable sets
[*] Uncountability of the real-number system
[*] Set algebra
[*] Countable collection of countable sets
[*] Exercises
[/LIST]
[*] Elements of Point Set Topology
[LIST]
[*] Introduction
[*] Euclidean space [itex]R^n[/itex]
[*] Open balls and open sets in [itex]R^n[/itex]
[*] The structure of open sets in [itex]R^1[/itex]
[*] Closed sets
[*] Adherent points. Accumulation points
[*] Closed sets and adherent points
[*] Bolzano-Weierstrass theorem
[*] The Cantor intersection theorem
[*] The Lindelof covering theorem
[*] The Heine-Borel covering theorem
[*] Compactness in [itex]R^n[/itex]
[*] Metric spaces
[*] Point set topology in metric spaces
[*] Compact subsets of a metric space
[*] Boundary of a set
[*] Exercises
[/LIST]
[*] Limits and Continuity
[LIST] 
[*] Introduction
[*] Convergent sequences in a metric space
[*] Cauchy sequences
[*] Complete metric spaces
[*] Limit of a function
[*] Limits of complex-valued functions
[*] Limits of vector-valued functions
[*] Continuous functions
[*] Continuity of composite functions
[*] Continuous complex-valued and vector-valued functions
[*] Examples of continuous functions
[*] Continuity and inverse images of open or closed sets
[*] Functions continuous on compact sets
[*] Topological mappings (homeomorphisms)
[*] Bolzano's theorem
[*] Connectedness
[*] Components of a metric space
[*] Arcwise connectedness
[*] Uniform continuity
[*] Uniform continuity and compact sets
[*] Fixed-point theorem for contractions
[*] Discontinuities of real-valued functions
[*] Monotonic functions
[*] Exercises
[/LIST]
[*] Derivatives
[LIST]
[*] Introduction
[*] Definition of derivative
[*] Derivatives and continuity
[*] Algebra of derivatives
[*] The chain rule
[*] One-sided derivatives and infinite derivatives
[*] Functions with nonzero derivative
[*] Zero derivatives and local extrema
[*] Rolle's theorem
[*] The Mean-Value Theorem for derivatives
[*] Intermediate-value theorem for derivatives
[*] Taylor's formula with remainder
[*] Derivatives of vector-valued functions
[*] Partial derivatives
[*] Differentiation of functions of a complex variable
[*] The Cauchy-Riemann equations
[*] Exercises
[/LIST]
[*] Functions of Bounded Variation and Rectifiable Curves
[LIST]
[*] Introduction
[*] Properties of monotonic functions
[*] Functions of bounded variation
[*] Total variation
[*] Additive property of total variation
[*] Total variation on [a,x] as a function of x
[*] Functions of bounded variation expressed as the difference of bounded functions
[*] Continuous functions of bounded variation
[*] Curves and paths
[*] Rectifiable paths and arc length
[*] Additive and continuity properties of arc length
[*] Equivalence of paths. Change of parameter
[*] Exercises
[/LIST]
[*] The Riemann-Stieltjes Integral
[LIST]
[*] Introduction
[*] Notation
[*] The definition of the Riemann-Stieltjes integral
[*] Linear properties
[*] Integration by parts
[*] Change of variable in a Riemann-Stieltjes integral
[*] Reduction to a Riemann integral
[*] Step functions as integrators
[*] Reduction of a Riemann-Stieltjes integral to a finite sum
[*] Euler's summation formula
[*] Monotonically increasing integrators. Upper and lower integrals
[*] Additive and linearity properties of upper and lower integrals
[*] Riemann's condition
[*] Comparison theorems
[*] Integrators of bounded variation
[*] Sufficient conditions for existence of Riemann-Stieltjes integrals
[*] Necessary conditions for existence of Riemann-Stieltjes integrals
[*] Mean Value Theorems for Riemann-Stieltjes integrals
[*] The integral as a function of the interval
[*] Second fundamental theorem of integral calculus
[*] Change of variable in a Riemann integral
[*] Second Mean-Value Theorem for Riemann integrals
[*] Riemann-Stieltjes integrals depending on a parameter
[*] Differentiation under the integral sign
[*] Interchanging the order of integration
[*] Lebesgue's criterion for existence of Riemann integrals
[*] Complex-valued Riemann-Stieltjes integrals
[*] Exercises
[/LIST]
[*] Infinite Series and Infinite Products
[LIST]
[*] Introduction
[*] Convergent and divergent sequences of complex numbers
[*] Limit superior and limit inferior of a real-valued sequence
[*] Monotonic sequences of real numbers
[*] Infinite series
[*] Inserting and removing parentheses
[*] Alternating series
[*] Absolute and conditional convergence
[*] Real and imaginary parts of a complex series
[*] Tests for convergence of series with positive terms
[*] The geometric series
[*] The integral test
[*] The big oh and little oh notation
[*] The ratio test and the root test
[*] Dirichlet's test and Abel's test
[*] Partial sums of the geometric series [itex]\sum z^n[/itex] on the unit circle |z|=1
[*] Rearrangements of series
[*] Riemann's theorem on conditionally convergent series
[*] Subseries
[*] Double sequences
[*] Double series
[*] Rearrangement theorem for double series
[*] A sufficient condition for equality of iterated series
[*] Multiplication of series
[*] Cesaro summability
[*] Infinite products
[*] Euler's product for the Riemann zeta function
[*] Exercises
[/LIST]
[*] Sequences of Functions
[LIST]
[*] Pointwise convergence of sequences of functions
[*] Examples of sequences of real-values functions
[*] Definition of uniform convergence
[*] Uniform concergence and continuity
[*] The Cauchy condition for uniform convergene
[*] Uniform convergence of infinite series of functions
[*] A space-filling curve
[*] Uniform convergence and Riemann-Stieltjes integration
[*] Nonuniformly convergent sequences that can be integrated term by
term
[*] Uniform convergence and differentiation
[*] Sufficient conditions for uniform convergence of a series
[*] Uniform convergence and double sequences
[*] Mean convergence
[*] Power series
[*] Multiplication of power series
[*] The substitution theorem
[*] Reciprocal of a power series
[*] Real power series
[*] The Taylor's series generated by a function
[*] Bernstein's theorem
[*] The binomial series
[*] Abel's limit theorem
[*] Tauber's theorem
[*] Exercises
[/LIST]
[*] The Lebesgue Integral
[LIST]
[*] Introduction
[*] The integral of a step function
[*] Monotonic sequences of step function
[*] Upper function and their integrals
[*] Riemann-integrable functions as eexamples of upper functions
[*] The class of Lebesgue-integrable functions on a general interval
[*] Basic properties of the Lebesgue integral
[*] Lebesgue integration and sets of measure zero
[*] The Levi monotone convergence theorems
[*] Applicatiom of Lebesgue's dominated convergence theorem
[*] Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals
[*] Improper Riemann integrals
[*] Measurable functions
[*] Continuity of functions defined by Lebesgue integrals
[*] Differentiation under the integral sign
[*] Interchanging the order of integration
[*] Measurable sets on the real line
[*] The Lebesgue integral over arbitrary subsets of R
[*] Lebesgue integrals of complex-valued functions
[*] Inner products and norms
[*] The set [itex]L^2(I)[/itex] of square-integrable functions
[*] The set [itex]L^2(I)[/itex] as a semimetric space
[*] A convergence theorem for series of functions in [itex]L^2(I)[/itex]
[*] The Riesz-Fischer theorem
[*] Exercises
[/LIST] 
[*] Fourier Series and Fourier Integrals
[LIST]
[*] Introduction
[*] Orthogonal systems of functions
[*] The theorem on best approximation
[*] The Fourier series of a function relative to an orthonormal system
[*] Properties of the Fourier coefficients
[*] The Riesz-Fischer theorem
[*] The convergence and representation problems for trigonometric series
[*] The Riemann-Lebesgue lemma
[*] The Dirichlet integrals
[*] An integral representation for the partial sums of a Fourier series
[*] Riemann's localization theorem
[*] Sufficient conditions for convergence of a Fourier series at a particular point
[*] Cesaro summability of Fourier series
[*] Consequence, of Fejer's theorem
[*] The Weierstrass approximation theorem
[*] Other forms of Fourier series
[*] The Fourier integral theorem
[*] The exponential form of the Fourier integral theorem
[*] Integral transforms
[*] Convolutions
[*] The convolution theorem for Fourier transforms
[*] The Poisson summation formula
[*] Exercises
[/LIST]
[*] Multivariable Differential Calculus
[LIST]
[*] Introduction
[*] The directional derivative
[*] Directional derivatives and continuity
[*] The total derivative
[*] The total derivative expressed in terms of partial derivatives
[*] An application to complex-values functions
[*] The matrix of linear function
[*] The Jacobian matrix
[*] The chain rule
[*] Matrix form of the chain rule
[*] The Mean-Value Theorem for differentiable functions
[*] A sufficient condition for differentiability
[*] A sufficient condition for equality of mixed partial derivatives
[*] Taylor's formula for functions from [itex]R^n[/itex] to [itex]R^1[/itex]
[*] Exercises
[/LIST]
[*] Implicit Functions and Extremum Problems
[LIST]
[*] Introduction
[*] Functions with nonzero Jacobian determinant
[*] The inverse function theorem
[*] The implicit function theorem
[*] Extrema of real-valued functions of one variable
[*] Extrema of real-valued functions of several variables
[*] Extremum problems with side conditions
[*] Exercises
[/LIST]
[*] Multiple Riemann Integrals
[LIST]
[*] Introduction
[*] The measure of a bounded interval in [itex]R^n[/itex]
[*] The Riemann integral of a bounded function defined on a compact interval in [itex]R^n[/itex]
[*] Sets of measure zero and Lebesgue's criterion for existence of a multiple Riemann integral
[*] Evaluation of a multiple integral by iterated integration
[*] Jordan-measurable sets in [itex]R^n[/itex]
[*] Multiple integration over Jordan-measurable sets
[*] Additive property of the Riemann integral
[*] Mean-Value Theorem for multiple integrals
[*] Exercises
[/LIST]
[*] Multiple Lebesgue Integrals
[LIST]
[*] Introduction
[*] Step functions and their integrals
[*] Upper functions end Lebesgue-integrable functions
[*] Measurable functions and measurable sets in [itex]R^n[/itex]
[*] Fubini's reduction theorem for double integrals
[*] Some properties of sets of measure zero
[*] Fubini's reduction theorem for double integrals
[*] The Tonelli-Hobson test for integrability
[*] Coordinate transformations
[*] The transformation formula for multiple integrals
[*] Proof of the transformation formula for linear coordinate transformations
[*] Proof of the transformation formula for the characteristic function of a compact cube
[*] Completion of the proof of the transformation formula
[*] Exercises
[/LIST]
[*] Cauchy's Theorem and the Residue Calculus
[LIST]
[*] Analytic functions
[*] Paths and curves in the complex plane
[*] Contour integrals
[*] The integral along a circular path as a function of the radius
[*] Cauchy's integral theorem for a circle
[*] Homotopic curves
[*] Invariance of contour integrals under homotopy
[*] General form of Cauchy's integral theorem
[*] Cauchy's integral formula
[*] The winding number of a circuit with respect to a point
[*] The unboundedness of the set of points with winding number zero
[*] Analytic functions defined by contour integrals
[*] Power-series expansions for analytic functions
[*] Cauchy's inequalities. Liouville's theorem
[*] Isolation of the zeros of an analytic function
[*] The identity theorem for analytic functions
[*] The maximum and minimum modulus of an analytic function
[*] The open mapping theorem
[*] Laurent expansions for functions analytic in an annulus
[*] Isolated singularities
[*] The residue of a function at an isolated singular point
[*] The Cauchy residue theorem
[*] Counting zeros and poles in a region
[*] Evaluation of real-valued integrals by means of residues
[*] Evaluation of Gauss's sum by residue calculus
[*] Application of the residue theorem to the inversion formula for Laplace transforms
[*] Conformal mappings
[*] Exercises
[/LIST]
[*] Index of Special Symbols
[*] Index
[/LIST]
 
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  • #10

Table of Contents:
Code:
[LIST]
[*] Preface
[*] General Topology
[LIST]
[*] Ordered Sets
[LIST]
[*] The axiom of choice, Zorn's lemma, Cantors's well-ordering principle; and 
their equivalence.
[*] Exercises.
[/LIST]
[*] Topology 
[LIST]
[*] Open and closed sets.
[*] Interior points and boundary. 
[*] Basis and subbasis for a topology. 
[*] Countability axioms. 
[*] Exercises. 
[/LIST]
[*] Convergence
[LIST]
[*] Nets and subnets.
[*] Convergence of nets.
[*] Accumulation points.
[*] Universal nets. 
[*] Exercises.
[/LIST]
[*] Continuity 
[LIST]
[*] Continuous functions.
[*] Open maps and homeomorphisms.
[*] Initial topology.
[*] Product topology.
[*] Final topology.
[*] Quotient topology.
[*] Exercises. 
[/LIST]
[*] Separation 
[LIST]
[*] Hausdorff spaces.
[*] Normal spaces.
[*] Urysohn's lemma.
[*] Tietze's extension theorem.
[*] Semicontinuity.
[*] Exercises.
[/LIST]
[*] Compactness 
[LIST]
[*] Equivalent conditions for compactness.
[*] Normality of compact Hausdorff spaces.
[*] Images of compact sets.
[*] Tychonoff's theorem.
[*] Compact subsets of R^n.
[*] The Tychonoff cube and metrization.
[*] Exercises.
[/LIST]
[*] Local Compactness 
[LIST]
[*] One-point compactification.
[*] Continuous functions vanishing at infinity.
[*] Normality of locally compact, a-compact spaces.
[*] Paracompactness.
[*] Partition of unity.
[/LIST]
[/LIST]
[*] Banach Spaces
[LIST]
[*] Normed Spaces 
[LIST]
[*] Normed spaces.
[*] Bounded operators.
[*] Quotient norm.
[*] Finite-dimensional spaces.
[*] Completion.
[*] Examples.
[*] Sum and product of normed spaces.
[*] Exercises. 
[/LIST]
[*] Category
[LIST]
[*] The Baire category theorem.
[*] The open mapping theorem.
[*] The closed graph theorem.
[*] The principle of uniform boundedness.
[*] Exercises. 
[/LIST]
[*] Dual Spaces 
[LIST]
[*] The Hahn-Banach extension theorem.
[*] Spaces in duality.
[*] Adjoint operator.
[*] Exercises.
[/LIST]
[*] Weak Topologies 
[LIST]
[*] Weak topology induced by seminorms.
[*] Weakly continuous functionals.
[*] The Hahn-Banach separation theorem.
[*] The weak* topology.
[*] w*-c1osed subspaces and their duality theory.
[*] Exercises.
[/LIST]
[*] w*-Compactness 
[LIST]
[*] Alaoglu's theorem.
[*] Krein-Milman's theorem.
[*] Examples of extremal sets.
[*] Extremal probability measures.
[*] Krein-Smulian's theorem.
[*] Vector-valued integration. 
[*] Exercises.
[/LIST]
[/LIST]
[*] Hilbert Spaces
[LIST]
[*] Inner Products
[LIST]
[*] Sesquilinear forms and inner products. 
[*] Polarization identities and the Cauchy*Schwarz inequality. 
[*] Parallellogram law. 
[*] Orthogonal sum. 
[*] Orthogonal complement. 
[*] Conjugate self-duality of Hilbert spaces. 
[*] Weak topology. 
[*] Orthonormal basis.
[*] Orthonormalization.
[*] Isomorphism of Hilbert spaces. 
[*] Exercises. 
[/LIST]
[*] Operators on Hilbert Space
[LIST]
[*] The correspondence between sesquilinear forms and operators. 
[*] Adjoint operator and involution in B(H).
[*] Invertibility, normality, and positivity in B(H).
[*] The square root.
[*] Projections and diagonalizable operators.
[*] Unitary operators and partial isometries.
[*] Polar decomposition.
[*] The Russo-Dye-Gardner theorem. 
[*] Numerical radius.
[*] Exercises. 
[/LIST]
[*] Compact Operators
[LIST]
[*] Equivalent characterizations of compact operators.
[*] The spectral theorem for normal, compact operators.
[*] Atkinson's theorem.
[*] Fredholm operators and index.
[*] Invariance properties of the index.
[*] Exercises. 
[/LIST]
[*] The Trace
[LIST]
[*] Definition and invariance properties of the trace.
[*] The trace class operators and the Hilbert-Schmidt operators.
[*] The dualities between B_0(H), B_1(H) and B(H).
[*] Fredholm equations.
[*] The Sturm-Liouville problem.
[*] Exercises. 
[/LIST]
[/LIST]
[*] Spectral Theory 
[LIST]
[*] Banach Algebras 
[LIST]
[*] Ideals and quotients. 
[*] Unit and approximate units. 
[*] Invertible elements. 
[*] C. Neumann's series. 
[*] Spectrum and spectral radius. 
[*] The spectral radius formula. 
[*] Mazur's theorem. 
[*] Exercises. 
[/LIST]
[*] The Gelfand Transform 
[LIST]
[*] Characters and maximal ideals.
[*] The Gelfand transform.
[*] Examples, including Fourier transforms.
[*] Exercises. 
[/LIST]
[*] Function Algebras
[LIST]
[*] The Stone-Weierstrass theorem. 
[*] Involution in Banach algebras. 
[*] C*-algebras.
[*] The characterization of commutative C*-algebras. 
[*] Stone-Cech compactification of Tychonoff spaces.  
[*] Exercises. 
[/LIST]
[*] The Spectral Theorem, I 
[LIST]
[*] Spectral theory with continuous function calculus.
[*] Spectrum versus eigenvalues. 
[*] Square root of a positive operator. 
[*] The absolute value of an operator. 
[*] Positive and negative parts of a self-adjoint operator. 
[*] Fuglede's theorem. 
[*] Regular equivalence of normal operators. 
[*] Exercises. 
[/LIST]
[*] The Spectral Theorem, II 
[LIST]
[*] Spectral theory with Borel function calculus.
[*] Spectral measures.
[*] Spectral projections and eigenvalues.
[*] Exercises. 
[/LIST]
[*] Operator Algebra 
[LIST]
[*] Strong and weak topology on B(H).
[*] Characterization of strongly/weakly continuous functionals.
[*] The double commutant theorem.
[*] Von Neumann algebras. 
[*] The \sigma-weak topology.
[*] The \sigma-weakly continuous functionals.
[*] The predual of a von Neumann algebra.
[*] Exercises. 
[/LIST]
[*] Maximal Commutative Algebras
[LIST]
[*] The condition U=U'.
[*] Cyclic and separating vectors.
[*] L^\infty(X) as multiplication operators.
[*] A measure-theoretic model for MACA's.
[*] Multiplicity-free operators.
[*] MACA's as a generalization of orthonormal bases.
[*] The spectral theorem revisited.
[*] Exercises. 
[/LIST]
[/LIST]
[*] Unbounded Operators 
[LIST]
[*] Domains, Extensions, and Graphs 
[LIST]
[*] Densely defined operators.
[*] The adjoint operator.
[*] Symmetric and self-adjoint operators. 
[*] The operator T*T.
[*] Semibounded operators.
[*] The Friedrichs extension.
[*] Examples. 
[/LIST]
[*] The Cayley Transform 
[LIST]
[*] The Cayley transform of a symmetric operator.
[*] The inverse transformation. 
[*] Defect indices.
[*] Affiliated operators.
[*] Spectrum of unbounded operators.
[/LIST]
[*] Unlimited Spectral Theory 
[LIST]
[*] Normal operators affiliated with a MACA.
[*] The multiplicity-free case.
[*] The spectral theorem for an unbounded, self-adjoint operator.
[*] Stone's theorem. 
[*] The polar decomposition.
[/LIST]
[/LIST]
[*] Integration Theory
[LIST] 
[*] Radon Integrals 
[LIST]
[*] Upper and lower integral.
[*] Daniell's extension theorem.
[*] The vector lattice L^1(X). 
[*] Lebesgue's theorems on monotone and dominated convergence. 
[*] Stieltjes integrals.
[/LIST]
[*] Measurability 
[LIST]
[*] Sequentially complete function classes.
[*] \sigma-rings and \sigma-algebras.
[*] Borel sets and functions.
[*] Measurable sets and functions.
[*] Integrability of measurable functions.
[/LIST] 
[*] Measures 
[LIST]
[*] Radon measures.
[*] Inner and outer regularity.
[*] The Riesz representation theorem. 
[*] Essential integral. 
[*] The \sigma-compact case.
[*] Extended integrability. 
[/LIST]
[*] L^p-spaces
[LIST]
[*] Null functions and the almost everywhere terminology.
[*] The HOlder and Minkowski inequalities.
[*] Egoroff's theorem.
[*] Lusin's theorem.
[*] The Riesz-Fischer theorem.
[*] Approximation by continuous functions.
[*] Complex spaces.
[*] Interpolation between L^p-spaces.
[/LIST]
[*] Duality Theory 
[LIST]
[*] \sigma-compactness and \sigma-finiteness.
[*] Absolute continuity.
[*] The Radon-Nikodym theorem.
[*] Radon charges.
[*] Total variation.
[*] The Jordan decomposition.
[*] The duality between L^p-spaces. 
[/LIST]
[*] Product Integrals 
[LIST]
[*] Product integral.
[*] Fubini's theorem.
[*] Tonelli's theorem.
[*] Locally compact groups.
[*] Uniqueness of the Haar integral.
[*] The modular function.
[*] The convolution algebras L^1(G) and M(G). 
[/LIST]
[/LIST]
[*] Bibliography 
[*] List of Symbols 
[*] Index
[/LIST]
 
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  • #11

Table of Contents:
Code:
[LIST]
[*] Foreword
[*] Preface
[*] The Genesis of Fourier Analysis
[LIST]
[*] The vibrating string
[LIST]
[*] Derivation of the wave equation 
[*] Solution to the wave equation 
[*] Example: the plucked string 
[/LIST]
[*] The heat equation
[LIST]
[*] Derivation of the heat equation 
[*] Steady-state heat equation in the disc
[/LIST]
[*] Exercises 
[*] Problem 
[/LIST]
[*] Basic Properties of Fourier Series
[LIST]
[*] Examples and formulation of the problem
[LIST]
[*] Main definitions and some examples
[/LIST]
[*] Uniqueness of Fourier series 
[*] Convolutions 
[*] Good kernels 
[*] Cesaro and Abel summability: applications to Fourier series
[LIST]
[*] Cesaro means and summation 
[*] Fejer's theorem 
[*] Abel means and summation 
[*] The Poisson kernel and Dirichlet's problem in the 
unit disc
[/LIST]
[*] Exercises 
[*] Problems
[/LIST]
[*] Convergence of Fourier Series
[LIST]
[*] Mean-square convergence of Fourier series
[LIST]
[*] Vector spaces and inner products 
[*] Proof of mean-square convergence
[/LIST]
[*] Return to pointwise convergence
[LIST]
[*] A local result 
[*] A continuous function with diverging Fourier series 
[/LIST] 
[*] Exercises 
[*] Problems 
[/LIST]
[*] Some Applications of Fourier Series
[LIST]
[*] The isoperimetric inequality 
[*] Weyl's equidistribution theorem 
[*] A continuous but nowhere differentiable function 
[*] The heat equation on the circle 
[*] Exercises 
[*] Problems 
[/LIST]
[*] The Fourier Transform on R
[LIST]
[*] Elementary theory of the Fourier transform
[LIST]
[*] Integration of functions on the real line
[*] Definition of the Fourier transform
[*] The Schwartz space
[*] The Fourier transform on S
[*] The Fourier inversion
[*] The Plancherel formula
[*] Extension to functions of moderate decrease 
[*] The Weierstrass approximation theorem
[/LIST]
[*] Applications to some partial differential equations
[LIST]
[*] The time-dependent heat equation on the real line
[*] The steady-state heat equation in the upper half-plane
[/LIST]
[*] The Poisson summation formula
[LIST]
[*] Theta and zeta functions
[*] Heat kernels
[*] Poisson kernels
[/LIST]
[*] The Heisenberg uncertainty principle
[*] Exercises
[*] Problems
[/LIST]
[*] The Fourier Transform on R^d
[LIST]
[*] Preliminaries
[LIST]
[*] Symmetries
[*] Integration on R^d
[/LIST]
[*] Elementary theory of the Fourier transform
[*] The wave equation in R^d x R
[LIST]
[*] Solution in terms of Fourier transforms
[*] The wave equation in R^3 x R
[*] The wave equation in R^2 x R: descent
[/LIST]
[*] Radial symmetry and Bessel functions
[*] The Radon transform and some of its applications 
[LIST]
[*] The X-ray transform in R^2
[*] The Radon transform in R^3
[*] A note about plane waves
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Finite Fourier Analysis
[LIST]
[*] Fourier analysis on Z(N)
[LIST]
[*] The group Z(N)
[*] Fourier inversion theorem and Plancherel identity on Z(N)
[*] The fast Fourier transform
[/LIST]
[*] Fourier analysis on finite abelian groups
[LIST]
[*] Abelian groups
[*] Characters
[*] The orthogonality relations
[*] Characters as a total family
[*] Fourier inversion and Plancherel formula
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Dirichlet's Theorem
[LIST]
[*] A little elementary number theory
[LIST]
[*] The fundamental theorem of arithmetic
[*] The infinitude of primes
[/LIST]
[*] Dirichlet's theorem
[LIST]
[*] Fourier analysis, Dirichlet characters, and reduction of the theorem
[*] Dirichlet L-functions
[/LIST]
[*] Proof of the theorem
[LIST]
[*] Logarithms 
[*] L-functions
[*] Non-vanishing of the L-function 
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Appendix: Integration
[LIST]
[*] Definition of the Riemann integral
[LIST]
[*] Basic properties
[*] Sets of measure zero and discontinuities of integrable functions
[/LIST]
[*] Multiple integrals
[LIST]
[*]  The Riemann integral in R^d
[*] Repeated integrals
[*] The change of variables formula
[*] Spherical coordinates
[/LIST]
[*] Improper integrals. Integration over R^d
[LIST]
[*] Integration of functions of moderate decrease
[*] Repeated integrals
[*] Spherical coordinates
[/LIST]
[/LIST]
[*] Notes and References
[*] Bibliography
[*] Symbol Glossary
[/LIST]
 
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  • #12

Table of Contents:
Code:
[LIST]
[*] Foreword
[*] Introduction
[LIST]
[*] Fourier series: completion
[*] Limits of continuous functions
[*] Length of curves
[*] Differentiation and integration
[*] The problem of measure
[/LIST]
[*] Measure Theory
[LIST]
[*] Preliminaries
[*] The exterior measure
[*] Measurable sets and the Lebesgue measure
[*] Measurable functions
[LIST]
[*] Definition and basic properties
[*] Approximation by simple functions or step functions
[*] Littlewoord's three principles
[/LIST]
[*] The Brunn-Minkowski inequality
[*] Exercises
[*] Problems
[/LIST]
[*] Integration Theory
[LIST]
[*] The Lebesgue integral: basic properties and convergence theorems
[*] The space L^1 of integrable functions
[*] Fubini's theorem
[LIST]
[*] Statement and proof of the theorem
[*] Applications of Fubini's theorem
[/LIST]
[*] A Fourier inversion formula
[*] Exercises
[*] Problems
[/LIST]
[*] Differentiation and Integration
[LIST]
[*] Differentiation of the integral
[LIST]
[*] The Hardy-Littlewood maximal function
[*] The Lebesgue differentiation theorem
[/LIST]
[*] Good kernels and approximations to the identity
[*] Differentiability of functions
[LIST]
[*] Functions of bounded variation
[*] Absolutely continuous functions
[*] Differentiability of jump functions
[/LIST]
[*] Rectifiable curves and the isoperimetric inequality
[LIST]
[*] Minkowski content of a curve
[*] Isoperimetric inequality
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Hilbert Spaces: An Introduction
[LIST]
[*] The Hilbert space L^2
[*] Hilbert spaces
[LIST]
[*] Orthogonality
[*] Unitary mappings
[*] Pre-Hilbert spaces
[/LIST]
[*] Fourier series and Fatou's theorem
[LIST]
[*] Fatou's theorem
[/LIST]
[*] Closed subspaces and orthogonal projections
[*] Linear transformations
[LIST]
[*] Linear functionals and the Riesz representation theorem
[*] Adjoints
[*] Examples
[/LIST]
[*] Compact operators
[*] Exercises
[*] Problems
[/LIST]
[*] Hilbert Spaces: Several Examples
[LIST]
[*] The Fourier transform on L^2
[*] The Hardy space of the upper half-plane
[*] Constant coefficient partial differential equations
[LIST]
[*] Weak solutions
[*] The main theorem and key estimate
[/LIST]
[*] The Dirichlet principle
[LIST]
[*] Harmonic functions
[*] The boundary value problem and Dirichlet's principle
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Abstract Measure and Integration Theory
[LIST]
[*] Abstract measure spaces
[LIST]
[*] Exterior measures and Caratheodory's theorem
[*] Metric exterior measures
[*] The extension theorems
[/LIST]
[*] Integration on a measure space
[*] Examples
[LIST]
[*] Product measures and a general Fubini theorem
[*] Integration formula for polar coordinates
[*] Borel measures on R and the Lebesgue-Stieltjes integral
[/LIST]
[*] Absolute continuity of measures
[LIST]
[*] Signed measures
[*] Absolute continuity
[/LIST]
[*] Ergodic theorems
[LIST]
[*] Mean ergodic theorem
[*] Maximal ergodic theorem
[*] Pointwise ergodic theorem
[*] Ergodic measure-preserving transformations
[/LIST]
[*] Appendix: the spectral theorem
[LIST]
[*] Statement of the theorem
[*] Positive operators
[*] Proof of the theorem
[*] Spectrum
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Hausdorff Measure and Fractals
[LIST]
[*] Hausdorff measure
[*] Hausdorff dimension
[LIST]
[*] Examples
[*] Self-similarity
[/LIST]
[*] Space-filling curves
[LIST]
[*] Quartic intervals and dyadic squares
[*] Dyadic correspondence
[*] Construction of the Peano mapping
[/LIST]
[*] Besicovitch sets and regularity
[LIST]
[*] The Radon transform
[*] Regularity of sets when d\geq 3
[*] Besicovitch sets have dimension 2
[*] Construction of a Besicovtich set
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Notes and References
[*] Bibliography
[*] Symbol Glossary
[*] Index
[/LIST]
 
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  • #13

Table of Contents:
Code:
[LIST]
[*] Notions from Set Theory
[LIST]
[*] Sets and elements. Subsets
[*] Operations on sets
[*] Functions
[*] Finite and infinite sets
[*] Problems
[/LIST]
[*] The Real Number System
[LIST]
[*] The field properties
[*] Order
[*] The least upper bound property
[*] The  existence of square roots
[*] Problems
[/LIST]
[*] Metric Spaces
[LIST]
[*] Definition of metric space. Examples
[*] Open and closed sets
[*] Convergent sequences
[*] Completeness
[*] Compactness
[*] Connectedness
[*] Problems
[/LIST]
[*] Continuous Functions
[LIST]
[*] Definition of continuity.  Examples
[*] Continuity and limits
[*] The continuity of rational operations. Functions with values in E^n
[*] Continuous functions on a compact metric space
[*] Continuous functions on a connected metric space
[*] Sequences of functions
[*] Problems
[/LIST]
[*] Differentiation
[LIST]
[*] The definition of derivative
[*] Rules of differentiation
[*] The mean value theorem
[*] Taylor's theorem
[*] Problems
[/LIST]
[*] Riemann Integration
[LIST]
[*] Definitions and examples
[*] Linearity and order properties of the integral
[*] Existence of the integral
[*] The fundamental theorem of calculus
[*] The logarithmic and exponential functions
[*] Problems
[/LIST]
[*] Interchange of Limit Operations
[LIST]
[*] Integration and differentiation of sequences of functions
[*] Infinite series
[*] Power series
[*] The trigonometric functions
[*] Differentiation under the integral sign
[*] Problems
[/LIST]
[*] The Method of Successive Approximations
[LIST]
[*] The fixed point theorem
[*] The simplest case of the implicit function theorem
[*] Existence and uniqueness theorems for ordinary differential equations
[*] Problems
[/LIST]
[*] Partial Differentiation
[LIST]
[*] Definitions and basic properties
[*] Higher derivatives
[*] The implicit function theorem
[*] Problems
[/LIST]
[*] Multiple Integrals
[LIST]
[*] Riemann integration on a closed interval in E^n. Examples and basic properties
[*] Existence of the integral. Integration on arbitrary subsets of E^n. Volume
[*] Iterated integrals
[*] Change of variables
[*] Problems
[/LIST]
[*] Suggestions for Further Reading
[*] Index
[/LIST]
 
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  • #14

Table of Contents:
Code:
[LIST]
[*] Preface
[*] Complex Numbers
[LIST]
[*] The Algebra  of Complex Numbers
[LIST]
[*] Arithmetic Operations
[*] Square Roots
[*] Justification
[*] Conjugation, Absolute Value
[*] Inequalities
[/LIST]
[*] The Geometric Representation of Complex Numbers
[LIST]
[*] Geometric Addition and Multiplication 
[*] The Binomial Equation
[*] Analytic Geometry
[*] The Spherical Representation
[/LIST]
[/LIST]
[*] Complex Functions
[LIST]
[*] Introduction to the Concept of Analytic Function
[LIST]
[*] Limits and Continuity 
[*] Analytic Functions 
[*] Polynomials 
[*] Rational Functions
[/LIST]
[*] Elementary Theory of Power Series
[LIST]
[*] Sequences 
[*] Series 
[*] Uniform Convergence 
[*] Power Series 
[*] Abel's Limit Theorem
[/LIST]
[*] The Exponential and Trigonometric Functions
[LIST]
[*] The Exponential 
[*] The Trigonometric Functions 
[*] The Periodicity. 
[*] The Logarithm
[/LIST]
[/LIST]
[*] Analytic Functions as Mappings
[LIST]
[*] Elementary Point Set Topology
[LIST]
[*] Sets and Elements 
[*] Metric Spaces 
[*] Connectedness 
[*] Compactness 
[*] Continuous Functions 
[*] Topological Spaces
[/LIST]
[*] Conformality
[LIST]
[*] Arcs and Closed Curves 
[*] Analytic Functions in Regions 
[*] Conformal Mapping 
[*] Length and Area
[/LIST]
[*] Linear Transformations
[LIST]
[*] The Linear Group 
[*] The Cross Ratio 
[*] Symmetry 
[*] Oriented Circles 
[*] Families of Circles
[/LIST]
[*] Elementary Conformal Mappings
[LIST]
[*] The  Use  of Level  Curves
[*] A Survey of Elementary Mappings
[*] Elementary Riemann Surfaces
[/LIST]
[/LIST]
[*] Complex Integration
[LIST]
[*] Fundamental Theorems
[LIST]
[*] Line Integrals 
[*] Rectifiable Arcs 
[*] Line Integrals as Functions of Arcs 
[*] Cauchy's Theorem for a Rectangle 
[*] Cauchy's Theorem in a Disk 
[/LIST]
[*] Cauchy's Integral Formula
[LIST]
[*] The Index of a Point with Respect to a Closed Curve 
[*] The Integral Formula 
[*] Higher Derivatives
[/LIST]
[*] Local Properties of Analytical Functions
[LIST]
[*] Removable Singularities. Taylor's Theorem 
[*] Zeros and Poles 
[*] The Local Mapping 
[*] The Maximum Principle
[/LIST]
[*] The General For  of Cauchy's Theorem
[LIST]
[*] Chains and Cycles 
[*] Simple Connectivity 
[*] Homology 
[*] The General Statement of Cauchy's Theorem 
[*] Proof of Cauchy's Theorem 
[*] Locally Exact Differentials 
[*] Multiply Connected  Regions
[/LIST]
[*] The Calculus of Residues
[LIST]
[*] The Residue Theorem
[*] The Argument Principle 
[*] Evaluation of Definite Integrals
[/LIST]
[*] Harmonic Functions
[LIST]
[*] Definition and Basic Properties 
[*] The Mean-value Property 
[*] Poisson's Formula 
[*] Schwarz's Theorem 
[*] The  Reflection Principle
[/LIST]
[/LIST]
[*] Series and Product Developments
[LIST]
[*]  Power Series Expansions
[LIST]
[*] Weierstrass's Theorem 
[*] The Taylor Series 
[*] The Laurent Series
[/LIST]
[*] Partial Fractions and Factorization
[LIST]
[*] Partial Fractions 
[*] Infinite Products 
[*] Canonical Products 
[*] The  Gamma Function 
[*] Stirling's Formula
[/LIST]
[*] Entire Functions
[LIST]
[*] Jensen's Formula 
[*] Hadamard's Theorem
[/LIST]
[*] The Riemann Zeta Function
[LIST]
[*] The  Product Development 
[*] Extension of \zeta(s) to the Whole Plane 
[*] The Functional Equation 
[*] The Zeros of the Zeta Function
[/LIST]
[*] Normal Families
[LIST]
[*] Equicontinuity 
[*] Normality and Compactness 
[*] Arzela's Theorem 
[*] Families of Analytic Functions 
[*] The Classical Definition 
[/LIST]
[/LIST]
[*] Conformal Mapping. Dirichlet's Problem
[LIST]
[*] The Riemann Mapping Theorem
[LIST]
[*] Statement and Proof 
[*] Boundary Behavior 
[*] Use of the Reflection Principle 
[*] Analytic Arcs
[/LIST]
[*] Conformal Mapping of Polygons
[LIST]
[*] The Behavior at an Angle 
[*] The Schwarz-Christoffel Formula 
[*] Mapping on a Rectangle 
[*] The Triangle Functions of Schwarz
[/LIST]
[*] A Closer Look at Harmonic Functions
[LIST]
[*] Functions with the Mean-value Property 
[*] Harnack's Principle
[/LIST]
[*] The Dirichlet Problem
[LIST]
[*] Subharmonic Functions 
[*] Solution of Dirichlet's Problem 
[/LIST]
[*] Canonical Mappings of Multiply Connected Regions 
[LIST]
[*] Harmonic Measures
[*] Green's Function
[*] Parallel Slit Regions
[/LIST]
[/LIST]
[*] Elliptic Functions
[LIST]
[*] Simply Periodic Functions
[LIST]
[*] Representation by Exponentials 
[*] The Fourier Development 
[*] Functions of Finite Order
[/LIST]
[*] Doubly Periodic Functions
[LIST]
[*] The Period Module 
[*] Unimodular Transformations 
[*] The Canonical Basis 
[*] General Properties of Elliptic Functions
[/LIST]
[*] The Weierstrass Theory
[LIST]
[*] The Weierstrass p-function 
[*] The Functions \zeta(z) and \sigma(z) 
[*] The Differential Equation 
[*] The Modular Function \lambda(\tau)
[*] The Conformal Mapping by \lambda(\tau)
[/LIST]
[/LIST]
[*] Global Analytic Functions
[LIST]
[*] Analytic Continuation
[LIST]
[*] The Weierstrass Theory
[*] Germs and Sheaves 
[*] Sections and Riemann Surfaces 
[*] Analytic Continuations along  Arcs 
[*] Homotopic Curves 
[*] The Monodromy Theorem 
[*] Branch Points
[/LIST]
[*] Algebraic Functions
[LIST]
[*] The Resultant of Two Polynomials 
[*] Definition and Properties of Algebraic Functions 
[*] Behavior at the  Critical Points
[/LIST]
[*] Picard's Theorem
[LIST]
[*] Lacunary Values
[/LIST]
[*] Linear Differential Equations
[LIST]
[*] Ordinary Points 
[*] Regular Singular Points 
[*] Solutions at Infinity 
[*] The Hypergeometric Differential Equation 
[*] Riemann's Point of View
[/LIST]
[/LIST]
[*] Index
[/LIST]
 
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  • #15

Table of Contents:
Code:
[LIST]
[*] Preface
[*] Introduction
[*] The Complex Plane and Elementary Functions
[LIST]
[*] Complex Numbers
[*] Polar Representation
[*] Stereographic Projection
[*] The Square and Square Root Functions
[*] The Exponential Function
[*] The Logarithm Function
[*] Power Functions and Phase Factors
[*] Trigonometric and Hyperbolic Functions
[/LIST]
[*] Analytic Functions
[LIST]
[*] Review of Basic Analysis
[*] Analytic Functions
[*] The Cauchy-Riemann Equations
[*] Inverse Mappings and the Jacobian
[*] Harmonic Functions
[*] Conformal Mappings
[*] Fractional Linear Transformations
[/LIST]
[*] Line Integrals and Harmonic Functions
[LIST]
[*] Line Integrals and Green's Theorem
[*] Independence of Path
[*] Harmonic Conjugates
[*] The Mean Value Property
[*] The Maximum Principle
[*] Applications to Fluid Dynamics
[*] Other Applications to Physics
[/LIST]
[*] Complex Integration and Analyticity
[LIST]
[*] Complex Line Integrals
[*] Fundamental Theorem of Calculus for Analytic Functions
[*] Cauchy's Theorem
[*] The Cauchy Integral Formula 
[*] Liouville's Theorem
[*] Morera's Theorem
[*] Goursat's Theorem
[*] Complex Notation and Pompeiu's Formula
[/LIST]
[*] Power Series
[LIST]
[*] Infinite Series
[*] Sequences and Series of Functions
[*] Power Series
[*] Power Series Expansion of an Analytic Function
[*] Power Series Expansion at Infinity
[*] Manipulation of Power Series
[*] The Zeros of an Analytic Function
[*] Analytic Continuation 
[/LIST]
[*] Laurent Series and Isolated Singularities
[LIST]
[*] The Laurent Decomposition
[*] Isolated Singularities of an Analytic Function 
[*] Isolated Singularity at Infinity
[*] Partial Fractions Decomposition
[*] Periodic Functions
[*] Fourier Series
[/LIST]
[*] The Residue Calculus
[LIST]
[*] The Residue Theorem
[*] Integrals Featuring Rational Functions
[*] Integrals of Trigonometric Functions
[*] Integrands with Branch Points
[*] Fractional Residues
[*] Principal Values
[*] Jordan's Lemma
[*] Exterior Domains
[/LIST]
[*] The Logarithmic Integral
[LIST]
[*] The Argument Principle
[*] Rouche's Theorem
[*] Hurwitz's Theorem 
[*] Open Mapping and Inverse Function Theorems
[*] Critical Points
[*] Winding Numbers
[*] The Jump Theorem for Cauchy Integrals
[*] Simply Connected Domains 
[/LIST]
[*] The Schwarz Lemma and Hyperbolic Geometry
[LIST]
[*] The Schwarz Lemma
[*] Conformal Self-Maps of the Unit Disk
[*] Hyperbolic Geometry
[/LIST]
[*] Harmonic Functions and the Reflection Principle
[LIST]
[*] The Poisson Integral Formula
[*] Characterization of Harmonic Functions
[*] The Schwarz Reflection Principle
[/LIST]
[*] Conformal Mapping
[LIST]
[*] Mappings to the Unit Disk and Upper Half-Plane
[*] The Riemann Mapping Theorem
[*] The Schwarz-Christoffel Formula
[*] Return to Fluid Dynamics
[*] Compactness of Families of Functions
[*] Proof of the Riemann Mapping Theorem
[/LIST]
[*] Compact Families of Meromorphic Functions 
[LIST]
[*] Marty's Theorem
[*] Theorems of Montel and Picard
[*] Julia Sets
[*] Connectedness of Julia Sets
[*] The Mandelbrot Set
[/LIST]
[*] Approximation Theorems
[LIST]
[*] Runge's Theorem
[*] The Mittag-Leffler Theorem
[*] Infinite Products
[*] The Weierstrass Product Theorem
[/LIST]
[*] Some Special Functions
[LIST]
[*] The Gamma Function
[*] Laplace Transforms
[*] The Zeta Function
[*] Dirichlet Series
[*] The Prime Number Theorem
[/LIST]
[*] The Dirichlet Problem
[LIST]
[*] Green's Formulae
[*] Subharmonic Functions
[*] Compactness of Families of Harmonic Functions
[*] The Perron Method
[*] The Riemann Mapping Theorem Revisited
[*] Green's Function for Domains with Analytic Boundary 
[*] Green's Function for General Domains
[/LIST]
[*] Riemann Surfaces
[LIST]
[*] Abstract Riemann Surfaces
[*] Harmonic Functions on a Riemann Surface
[*] Green's Function of a Surface
[*] Symmetry of Green's Function
[*] Bipolar Green's Function
[*] The Uniformization Theorem
[*] Covering Surfaces 
[/LIST]
[*] Hints and Solutions for Selected Exercises
[*] References
[*] List of Symbols
[*] Index 
[/LIST]
 
Last edited by a moderator:
  • #16
  • Author: Gerald B. Folland
  • Title: Fourier Analysis and Its Applications (Pure and Applied Undergraduate Texts) [Hardcover]
  • Amazon Link: https://www.amazon.com/dp/0821847902/?tag=pfamazon01-20
  • Prerequisities: Calculus (including multivariable), Differential equations,
  • Level: Undergraduate, intermediate to upper level; Graduate, introductory

Table of Contents
Code:
1 Overture
1.1 Some equations of mathematical physics
1.2 Linear differential operators
1.3 Separation of variable

2 Fourier Series
2.1 The Fourier series of a periodic function
2.2 A convergence theorem
2.3 Derivatives, integrals, and uniform convergence
2.4 Fourier series on interval
2.5 Some applications
2.6 Further remarks on Fourier series

3 Orthogonal Sets of Functions
3.1 Vectors and inner products
3.2 Functions and inner products
3.3 Convergence and completeness
3.4 More about L2 spaces; the dominated convergence theorem
3.5 Regular Sturm-Liouville problems
3.6 Singular Sturm-Liouville problems

4. Some Boundary Value Problems
4.1 Some useful techniques
4.2 One-dimensional heat flow
4.3 One-dimensional wave motion
4.4 The Dirichlet problem
4.5 Multiple Fourier series and applications

5 Bessel Functions
5.1 Solutions of Bessel's equation
5.2 Bessel function identities
5.3 Asymptotics and zeros of Bessel functions
5.4 Orthogonal sets of Bessel functions
5.5 Applications of Bessel functions
5.6 Variants of Bessel functions

6 Orthogonal Polynomials
6.1 Introduction
6.2 Legendre polynomials
6.3 Spherical coordinates and Legendre functions
6.4 Hermite polynomials
6.5 Laguerre polynomials
6.6 Other orthogonal bases

7 The Fourier Transform
7.1 Convolutions
7.2 The Fourier Transform
7.3 Some applications
7.4 Fourier transforms and Sturm-Liouville problems
7.5 Multivariable convolutions and Fourier transforms
7.6 Transforms related to the Fourier transform

The Laplace Transform
8.1 The Laplace Transform
8.2 The inversion formula
8.3 Applications: Ordinary differential equations
8.4 Applications: Partial differential equations
8.5 Applications: Integral equations
8.6 Asymptotics of Laplace transforms

9 Generalized Functions
9.1 Distributions
9.2 Convergence, convolution, and approximation
9.3 More examples: Periodic distributions and finite parts
9.4 Tempered distributions and Fourier transforms
9.5 Weak solutions of differential equations

10 Green's Functions
10.1 Green's functions for ordinary differential operators
10.2 Green's functions for partial differential operators
10.3 Green's functions and regular Sturm-Liouville problems
10.4 Green's functions and singular Sturm-Liouville problems

Appendices
1 Some physical derivations
2 Summary of complex variable theory
3 The gamma function
4 Calculations in polar coordinates
5 The fundamental theorem of ordinary differential equations

Answers to the Exercises
References
Index of Symbols
Index

Undergraduate and graduate students interested in studying the Fourier transform.

This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.

Hardcover: 433 pages
Publisher: American Mathematical Society (January 13, 2009)

http://www.ams.org/publications/authors/books/postpub/amstext-4
 
Last edited by a moderator:
  • #17
Hi guys,

What is your opinion on books below ? Could you please grade those, that you've read ? Many thanks.

Analysis in Vector Spaces - Mustafa A. Akcoglu, Paul F.A. Bartha, Dzung Minh Ha
https://www.amazon.com/dp/0470486775/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Principles of Functional Analysis - Martin Schechter
https://www.amazon.com/dp/0821828959/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Applied Functional Analysis: Applications to Mathematical Physics - Eberhard Zeidler
https://www.amazon.com/dp/0387944427/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Advanced Calculus, Third Edition - R. Creighton Buck
https://www.amazon.com/dp/1577663020/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Advanced Calculus [Paperback] - Angus E. Taylor, W. Robert Mann
https://www.amazon.com/dp/0471025666/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Analysis I [Paperback] - Herbert Amann , Joachim Escher , Gary Brookfield (Translator)
https://www.amazon.com/dp/B00868DGZE/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Analysis II [Paperback] - Herbert Amann , Joachim Escher
https://www.amazon.com/dp/3764374721/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Analysis III [Paperback] Herbert Amann , Joachim Escher Joachim Escher
https://www.amazon.com/dp/3764374799/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Calculus of Vector Functions - Richard E. Williamson , Richard H. Crowell , Hale F. Trotter
https://www.amazon.com/dp/013112367X/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Basic Real Analysis [Hardcover] - Houshang H. Sohrab
https://www.amazon.com/dp/0817642110/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Mathematical Analysis - S. C. Malik , Savita Arora
https://www.amazon.com/gp/product/1906574111/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Advanced Calculus for Applications, Second Edition [Paperback] - Francis B. Hildebrand
https://www.amazon.com/dp/0130111899/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Advanced Calculus for Engineers [Paperback] - Francis Begnaud Hildebrand
https://www.amazon.com/dp/1614273987/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Functional Analysis [Hardcover] - Walter Rudin
https://www.amazon.com/dp/0070542368/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Advanced Level Mathematics (Pure and Applied) [Paperback] Clement John Tranter
https://www.amazon.com/gp/product/0340242027/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Analysis I: Convergence, Elementary functions (Universitext) [Paperback] - Roger Godement
https://www.amazon.com/dp/3540059237/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions (Universitext) [Paperback] - Roger Godement
https://www.amazon.com/dp/3540209212/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Advanced calculus;: Problems and applications to science and engineering [Hardcover] - Hugo Rossi
https://www.amazon.com/gp/product/B0006C0C46/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Calculus with Analytic Geometry (Second Edition) [Hardcover] - John F. Randolph
https://www.amazon.com/dp/B000H5L6MY/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

A First Course in Mathematical Analysis [Paperback] - David Alexander Brannan
https://www.amazon.com/dp/0521684242/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Calculus and Linear Algebra: An Integrated Approach (Saunders mathematics books) [Hardcover] - Mary R. Embry
https://www.amazon.com/dp/0721633706/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):

Johnson and Kiokemeister's Calculus With Analytic Geometry [Hardcover] - Richard E. Johnson
https://www.amazon.com/dp/0205059171/?tag=pfamazon01-20
Your comment (grade, alternatives to this book and why):
 
  • #18
(Apologies, I could not think of a better title - also not sure if this is the correct subforum - also, I know that this is not a German regional forum, but the book was in English, so I thought may be the international community can help?).

Two days ago, I was in the Library at Köln (Those who know it, knowes that the books are not ordered by subject - but by date of buying, and one has to do catalogue research). and I was cassually browsing the racks for things that might interest me. Now I did find a book.

All what I remember is, it was a book of Non-linear analysis of time series. The author, at the beginning of some chapter (either 3,4 or 5 - or may be I am wrong), argues that in contrast to the previous chapters where he presented interpolation (e.g. B-Splines) and nearest neighbor technics, in the present chapter he will present technics that also make assumptions on the underlying dynamic process that generate the time series.

The actual sentences were like "it thus seems to be a good idea [...] to account for the underlying processes that generate the time serise".

However, afterwards, I found this interesting. So I wanted to read more, but I had to come back to Bonn - thus I wanted to check the book out, by had some fines and all - thus I just kept the book there and thought I will be back later.

Problem is, as I was rushing back to catch the train to Bonn, I forgot to right down the name and author of the book. Going through the search result returned by the catalouge does not seem to lead me to the book I was looking for - the book feels like as if vanished.

Can anyone help be by telling me the name of a book where the author argues as above?

Thanks a Lot.
 

1. What is time series analysis?

Time series analysis is a statistical method used to analyze data that is collected over a period of time. It involves identifying patterns or trends in the data and making predictions about future values based on those patterns.

2. Why is time series analysis important?

Time series analysis allows us to understand and make predictions about complex data sets, such as stock prices, weather patterns, and economic indicators. It can be used to identify trends, detect anomalies, and make informed decisions based on historical data.

3. What are some common techniques used in time series analysis?

Some common techniques used in time series analysis include moving averages, exponential smoothing, autoregressive integrated moving average (ARIMA) models, and seasonal decomposition. Other advanced techniques may include machine learning algorithms and neural networks.

4. What are some real-world applications of time series analysis?

Time series analysis is widely used in various industries, such as finance, economics, marketing, and healthcare. It can be used to forecast sales, predict stock market trends, monitor economic indicators, and analyze patient data for disease outbreaks.

5. Where can I find a good text on time series analysis?

There are many books and online resources available for time series analysis, but some popular texts include "Time Series Analysis and Its Applications" by Shumway and Stoffer, "Introductory Time Series with R" by Cowpertwait and Metcalfe, and "Forecasting: Principles and Practice" by Hyndman and Athanasopoulos. It's important to find a text that aligns with your level of expertise and specific interests in time series analysis.

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