Fourth Stage


Complex Analysis

Units : 8
Lecturer : Haveen J. Ahmed

Description : Course overview:
In first year you studied real-valued functions of a real variable. Real functions vary enormously in how nicely behaved they are in terms of properties like continuity, differentiability and so forth. The library of standard functions that we study, such as rational functions, trigonometric functions, and hyperbolic functions, are all extremely well-behaved; much better even than a 'typical' infinitely differentiable real function.
To understand why, it turns out that one should look at complex functions of a complex variable. Differentiability for these functions, although having an identical looking definition, is a much stronger property than in the real case. The behaviour of complex functions which are differentiable on a reasonable set is very restricted. In a certain sense, all such functions are almost polynomials and this explains many of the properties of our standard real functions. Studying these complex functions requires us to find new ways of picturing
functions. The graphical methods of real variable calculus are of limited use, but geometric reasoning will be just as important.
An important concept throughout the course will be that of a path integral. Here, instead of integrating a function over an interval [a,b], we integrate functions over a curve in the plane. Many of our main theorems will concern how one can evaluate these path integrals with doing any integration! The powerful integral formulas due to Cauchy and Goursat have lots of lovely consequences, including enabling us to do many real integrals for which our real valued methods fail.

Course objective:

This course is aimed to provide an introduction to the theories for functions of a complex variable. It begins with the exploration of the algebraic, geometric and topological structures of the complex number field. The concepts of analyticity, Cauchy-Riemann relations and harmonic functions are then introduced. The notion of the Riemann sheet is presented to help student visualize multi-valued complex functions. Complex integration and complex power series are presented. We then discuss the classification of isolated singularities and examine the theory and illustrate the applications of the calculus of residues in the evaluation of integrals. Students will be equipped with the understanding of the fundamental concepts of
complex variable theory. In particular, students will acquire the skill of contour integration to evaluate complicated real integrals via residue calculus. The prerequisites are some knowledge of calculus (up to line integrals and Green’s theorem), and some basic familiarity with differential equations would be useful.

Student learning outcome:
Students should be able to demonstrate their understanding of the basic knowledge in the theory of complex variables and applications in other subjects through appropriate homework assignment and examinations. Also they have to be able to competently solve problems using mathematical tools and to apply calculus techniques to solve problems.

The Topics:
Chapter (1): Complex numbers
Definition. Algebraic Properties. Cartesian Coordinates. The triangle Inequality. Polar Coordinates. Power and roots. Regions in the complex plane.
Chapter (2): Analytic functions
Functions of complex variables. Mappings. Limits. Theorems on Limits. Continuity. Derivatives. Differentiation Formulas. The Cauchy-Riemann Equations. Sufficient Conditions. The Cauchy-Riemann Equations in Polar Form. Analytic Functions. Harmonic Functions.
Chapter (3): Elementary Functions
Exponential functions. Logarithmic functions. Trigonometric and Hyperbolic functions. Inverse Trigonometric and Hyperbolic Functions.
Chapter (4): Mapping by Elementary Functions
The Linear Transformation. Transformation by some functions.
Chapter (5): Integrals
Definite Integrals. Contours. Line Integrals. The Cauchy-Goursat Theorem. Simply and Multiply Connected Dommains. The Cauchy-Goursat Formula. Derivatives of Analytic Functions. Liouville's Theorem and The Fundamental Theorem of Algebra.
Chapter (6): Sequences and Series
Converges of Sequences and Series. Taylor and Laurent Series. Absolute and Uniform Convergence of Power Series. Zeros of Analytic Functions and Singularities.
Chapter (7): Residue and Poles
Residues. The Residue Theorem. The Principal Part of a Function. Poles. Residues at Poles. Definite Integral of Trigonometric Functions. Evaluation of Improper Real Integrals. Integration Around a Branch Point.

Course Reading List and References:
- James Ward Broun and Ruel V. Churchill "Complex variables and Applications".
- H. R. Chilingworth "Complex Variables".
- E. B. Saff and A. D. Snider "Fundamentals of Complex Analysis". R. P. Boas "Invitation to Complex Variables".

Functional Analysis

Units : 4
Lecturer : Dr. Nareen Sabih Muhemed

Description : Course overview:
Functional analysis is generally regarded as an activity whereby functions are broken down into sub-functions, and relationships between sub-functions (e.g. sequence, concurrency, control flow, item flow, logical branching, looping, iteration, replication, etc.). Therefore, the breakdown (decomposition) of a function describes how that function is to be accomplished (performed). Functional analysis has two major applications.
The first is as a tool for CAPTURE and VALIDATION of requirements. That is, functional analysis is a technique used WITHIN requirements analysis. Anybody who has ever developed a use case, or written down how something is to be used for a particular use, has used functional analysis in this application. There are, of course, much more robust ways of using functional
analysis as a requirements analysis tool than the traditional IT use case approach. The second major application of functional analysis is as a design tool, a LOGICAL DESIGN tool. In this application, requirements level functions are broken down into solution level functions. Take the “Conduct the Olympic Games (a requirements level function)” system. When we define functions such as “design the stadium”, “build the stadium”, “obtain certificate of occupancy”, “conduct event trial”, etc., we are DECIDING UPON, we are CREATING, solution level functions. It is exactly the same with technologybased solutions. As soon as we decide that the function “design the stadium”is to be performed by the “stadium contractor” object, we have CREATED a

functional requirement on that object. That is the design application of functional analysis.

Course objective:
we will cover a large amount of material. This should give you some warning of the fact that this will not be a course to relax at; though this does not mean that you will not be able to enjoy it. Functional Analysis is not only a beautiful subject but it is also very useful and powerful in applied mathematics and theoretical physics. It is also basic for the understanding and development of very many other mathematical theories like the Theory of Partial Differential
Equations and Operator Theory.
Do not be fooled by the fact things start slow. This is the kind of course where things keep on building up continuously, with new things appearing rather often. In a mathematically sounding language, this course is "locally easy" but "globally hard". That means that if you keep up to date with the lectures, read my notes regularly and do the exercises, you should not have any problem and might even be able to enjoy this course. Otherwise, you will soon find yourself in deep trouble.

Student learning outcome:
A student who completes this module successfully should be able to:
- Define Banach space and Hilbert space ,
- Knew the deferent between functional and real analysis .
- Knew many important theorem in mathematics such as ,closed graph and open mapping theorem.
- Study Master, PhD because students will be obtained what is needed for them.

The Topics:
1. Vector spaces, linear combinations, linear dependence, linear independence, span, basis,
dimension, subspace of a vector space, further properties of vector spaces, theorems and examples.
2. Normed space, Banach space, convergence of sequence and series in a normed space, finite
dimensional normed space, subspaces, equivalent norms, the product of Banach spaces, isomorphism of a normed space, theorems and examples.
3. Linear operators, properties of linear operators , the null of the operators, bounded and continuous linear operators with applications, inverse of operators, the graph of the operators,
homeomorphism operators, linear operators on a finite dimensional normed space, theorems and examples.
4. Linear functional, bounded and continuous linear functional with applications, linear functional on a finite dimensional normed space, theorems and examples.
5. Fundamental theorems for normed and Banach spaces Zorn's lemma, Hahn Banach theorem, convergent of sequences of operators, open mapping theorem, closed graph theorem, partially ordered set, reflexive spaces, the dual space of a normed space reflexivity ona finite dimensional normed space and in a Hilbert space, theorems and examples.
6. Inner product (pre-Hilbert) space, Hilbert space. Convergence of a sequence in a pre-Hilbert space, subspaces of a Hilbert space, orthogonal complement, direct sum. Orthogonal  projection, orthonormal sets and sequences, Bessel inequality, Hilbert adjoint operators, self-adjoint operators, unitary operators, normal operators, self adjoint of product, sequence of self-adjoint operators, theorems and examples.

Course Reading List and References:
1. Introductory Functional Analysis with application, Erwin Kreyzice.
2. Element of Functional Analysis,2nd Edition I.J.Madox.
3. Introduction to Functional Analysis, Angus E. Taylor.
4. Functional Analysis/Terry J. Morrison

General Topology

Units : 8
Lecturer : Dr. Alias Barakat Khalaf

Description : Course overview:
The teaching of topology affords the instructor an opportunity not only to impart necessary mathematical content, but also to expose the student to both rigor and abstraction. This text, is designed to emphasize the value of careful presentations of proofs and to show the power of abstraction. Since the axiomatic method is fundamental to mathematics, the student should become acquainted with it as early as possible. This course present various amounts of material to students with quite varying backgrounds. The only prerequisite is the study of some analysis; even the traditional "Advanced Calculus" should be sufficient. It should be particularly noted that no part of this course requires the material sometimes labeled "Modern Algebra." Since it is possible to reach this level with no detailed knowledge of set theory. Students may need merely some review and practice in set theory. Well-prepared students may be started with the fundamentals of general topology in logical order progressing from the most general case of a topological space to the restrictive case of a complete metric space. Including the material introduced in the problems, there is more than enough material in the entire course for a two-semester course. Students who must start with the introductory chapters will cover correspondingly less of this additional material. The proofs will be given in
considerable detail so that they may serve as models for the student to emulate. Only after he has had experience in giving such detailed proofs is the student prepared to omit the "obvious" steps. Although no figures are given, the teacher should encourage the student to make diagrams to help his visualization. The student must be cautioned, however, that a picture is not a proof, which is the main reason for omitting them here. On the other hand, a topological spaces and metric spaces are defined and a brief treatment of Euclidean space is given. The motivation for the definition of a topology is based on the notion of open sets in basic calculus. The need for the definition of Hausdorff space is shown by stressing that the concept is essential to prove that the limit of a convergent sequence is unique. Continuity and homeomorphism are presented and based on the familiar concept of continuity in calculus. Product spaces will be discuss, we treat connectedness and consider the special case of connectedness on the real line, leading to the proof of the Intermediate Value Theorem. Different forms of compactness are treated , where we try to make the student appreciate that compactness is the vehicle that takes us from the infinite to the finite. Using compactness in the space of real numbers, we prove some important theorems from calculus.

Course objective:
The course provide a mathematically rigorous introduction to metric spaces, Topological spaces, product topology, Connected spaces, compact spaces, Separation axioms and complete metric topology. I have followed the principle that the material should be as clear and as intuitive as possible. A major aim of this course is to teach students to understand the concepts and the methods of proving of topological spaces, Connectedness, Compactness and separation. One way to describe the subject of Topology is to say that it is qualitative geometry. The idea is that if one geometric object can be continuously transformed into another, then the two objects are to be viewed as being topologically the same. For example, a circle and a square are topologically equivalent. Physically, a rubber band can be stretched into the form of either a circle or a square, as well as many other shapes which are also viewed as being topologically equivalent.

Student learning outcome:
Although this material is a prerequisite for the study of many fields of mathematics, here it
is presented to be studied for its own sake. It is hoped that the student finishing this text will choose to continue to study and do research in the field, and the number of references has been added for his benefit. The student is supposed to learn most of the mathematical concepts in pure mathematics that he studied during the previous three years.

The Topics:
Topology Of The R e a l Line And Metric Spaces
-Real line. Open sets. Accumulation points. Bolzano-W eierstrass theorem. Closed sets.  Sequences.Convergent sequences. Cauchy sequences.
Topological Spaces:
-Definitions,Topological spaces. Accumulation points. Closed sets. Closure of a set. Interior, exterior, boundary. Neighborhoods and neighborhood systems. Convergent sequences. Coarser and finer topologies. Subspaces, relative topologies. Equivalent definitions of topologies.
Bases And Subbases
-Base for a topology. Subbases. Topologies generated by classes of sets. Local bases.
Continuity And Topological Equivalence
Continuous functions. Continuous functions and arbitrary closeness. Continuity at a point. Sequential continuity at a point. Open and closed functions. Homeomorphic spaces.  Topological properties.
Separation Axioms
-Ti-spaces. Hausdorff spaces. Regular spaces. Normal spaces. Urysohn's lemma and  metrization theorem. Functions that separate points.
-Covers. Compact sets. Subsets of compact spaces. Finite intersection property. Compactness and Hausdorff spaces. Sequentially compact sets. Countably compact sets. Locally compact spaces. Compactification.
Product Spaces
-Product topology. Base for a finite product topology. Defining subbase and defining base for the product topology

Course Reading List and References:

Graph Theory

Units : 6
Lecturer : Nechirvan Badal Ibrahim

Description : Course overview:
This course focuses on the mathematical theory of graphs; a few applications and algorithms will be discussed. Topics include trees, connectivity, Eulerian and Hamiltonian graphs, edge and vertex colorings, Wiener Polynomial , planar graphs, and directed graphs. An advanced topic completes the course. Familiarity with linear algebra and basic counting methods such as binomial coefficients is assumed. Comfort with reading and writing mathematical proofs is required. The overall goal of the course is for students to learn the main concepts of Graph Theory and to be able to be able to write rigorous proofs involving these concepts.

Course objective:
This course aims are twofold. First, to discuss some of the major results of graph theory, and to provide an introduction to the language, methods and terminology of the subject. Second, to emphasise various approaches (algorithmic, probabilistic, etc) that have proved fruitful in  odern graph theory: these modes of thinking about the subject have also proved successful in  ther areas of mathematics, and we hope that students will find the techniques learnt in this  ourse to be useful in other areas of mathematics aims, and the understanding of fundamental definitions and properties of graphs, the ability to read and write rigorous mathematical proofs involving graphs, recognition of the numerous applications of graph theory in computer science and engineering.

Student learning outcome:
A student who masters of the content of this course will be qualified to:
- Continue with the study of special topics and to apply graph theory to other fields.
- Present the various topics in the theory of graphs in a logical order.
- Indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results.
- The course can thus be used as a text in the tradition of the more method, with the student
gaining mathematical power by being encouraged to prove all theorems stated without proof.
- Study Master, PhD because students will be obtained what is needed for them.

The Topics:
CH.1: Basic Result
-The Konigsberg Bridge problem, Basic Concepts, Degree of Vertices, Subgraphs, Path and Connectedness, Distance, The Transmission, Digraphs, Line graph, The Adjacency Matrix, Operations on Graphs.
CH. 2: Degree Sequences
-Degree Sequences of Graph, Degree of Singularity of Graph.
CH.3:Connectivity of Graphs
-Cut Vertex, Bridges and Blocks, Bipartite graphs and trees, Connectivity.
CH.4:Tours and matchings
-Eulerian graphs, Hamiltonian graphs Matchings.
CH.5: Colorability
-Vertex Colorings, Edge Coloring of Graphs, Homomorphism, The Chromatic polynomial, Connectivity, Vertex Connectivity and Edge Connectivity
CH.6: Graphs and surfaces
-Planar Graphs, Coloring planar Graphs, Genus of a Graphs, Dual graphs and Kuratowiski’s theorem.
CH.7: Wiener Polynomials
CH.8: Flow in Network

Course Reading List and References:
[1] R. Balakrishnan and K. Ranganathan; A Textbook of Graph Theory, SpringerVerlag, Inc., New York, 2000.
[2] L. W. Beineke and R. J. Wilson; Selected Topics in Graph Theory, Academic Press, Inc., London, 1978.
[3] F. Harary; Graph Theory, Addison-Wesley, Reading, MA., 1969.


Units : 4
Lecturer : Mardeen Sh .Taher

Description : Course overview:
Optimization is central to any problem involving decision making, whether in engineering or in economics. The task of decision making entails choosing between various alternatives. This choice is governed by our desire to make the "best" decision. The measure of goodness of the alternatives is described by an objective function or performance index. Optimization theory and methods deal with selecting the best alternative in the sense of the given objective  function.
The area of optimization has received enormous attention in recent years, primarily because of the rapid progress in computer technology, including the development and availability of  user-friendly software, high-speed and parallel processors, and artificialneural networks. A clear example of this phenomenon is the wide accessibility of optimization software tools such as the optimization Toolbox of MATLAB1 and the many other commercial software packages .

Mathematical optimization provides a unifying framework for studying issues of rational decision making , optimal design ,effective resource allocation and economic efficiency . It is therefore a central methodology of many business-related disciplines , including operations research , marketing , accounting ,economics , game theory and finance . In many of these disciplines, a solid background in optimization theory is essential for doing research. This course provides a rigorous introduction to the fundamental theory of optimization. It examines optimization theory in tow primary settings: in part one we consider unconstrained optimization problems . we first define some basic definitions and discuss some theoretical foundations of set constrained and unconstrained optimization in one dimension and malty dimension, including necessary and sufficient conditions for minimizers and maximizers .This is followed by a treatment of various iterative optimization algorithms , together with their properties .part, two deals with linear programming problems ,which form an important class of constrained optimization problems .The of the course is to provide students with a foundation sufficient to use basic optimization in their own research work and/or pursue more specialized studies involving optimization theory.

Course objective:
- Introduce students to numerical methods for solving smooth nonlinear optimization problems and to explore the theoretical background behind these methods.
- Be sufficiently familiar with a range of powerful numerical methods for solving nonlinear optimization problems;
- understand the theoretical background behind each of the methods including motivation, development, restrictions, advantages and disadvantages, and implementation;
- apply properly the resulting algorithms to solving practical optimization problems.

Student learning outcome:
1-Define and understand the basic definitions such as optimization, constrained and unconstrained optimization, Hessian matrix, Convex Sets and Convex Functions , exact and inexact line search.
2- Find the minimizer of the continuous function (one dimensional and ( MultiDimensional).
3- How to find the inverse Hessian matrix in different ways,
4- Differences between the different ways in one dimensional and ( MultiDimensional) , the advantage and disadvantage for each one of them.

The Topics:
-Unconstrained Optimization (Introduction and Background: Linear Algebra and Analysis. - Convex Sets and Convex Functions. -First Order Necessary Conditions, Second Order sufficiency Conditions.
-One-Dimensional Search Methods
1- Golden Section Search
2- Newton's Method
3- Secant Method
4- Cubic Interpolation
5- Quadratic Interpolation
-Multi-Dimensional Search Methods ,Direct Search Method ,Gradient Methods
1- Steepest Descent
2- Conjugate Gradient
3- Quasi-Newton Methods
a- Symmetric Rank One Method
b- Dvidon –Fletcher –Powell Method
c- Broyden, Fletcher –Goldfard –Shanno Method

Course Reading List and References:
1- Edwin K. P, Chony and Stanislaw H. Zak ,( 2001), An Introduction To Optimization.
2- David G. Luenberger, (1972), Introduction to Linear and Non-linear programming.
3- Practical Methods of Optimization, Fletcher R., John Wiley, 2000.

Research Project

Units : 4

Description :

No available

Special Functions

Units : 4
Lecturer : Dr. Shayma Adil Murad

Description : Course overview:
Special Functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications. There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. Many special functions appear as solutions of differential equations or integrals of elementary functions. In mathematics, several functions or groups of functions are important enough to deserve their own names. There is a large theory of special functions which developed out of statistics and mathematical physics. Special functions are just that : specialized functions beyond the familiar trigonometric or exponential functions. The one studied (hypergeometric functions, orthogonal polynomial, and so on) arise very naturally in areas of analysis, number theory, Lie groups, and combinatory. Very detailed information is often available. After successful completion of this course, students will be able to develop key concepts in Special functions.

Course objective:

The student will be able to:
- Understand the basic use for both the Gamma and Beta functions.
- Solve the Legendre differential equation by series method and find the conditions necessary for a polynomial solution.
- Derive and apply the generating function and recurrence relations for laguerre , hermite and Bessel's Polynomials
- Employ the orthogonality relation of Legendre polynomials to develop functions as series of such polynomials. Also Understand the basic use for modified Bessle's functions, moreover derive and apply the generating function and recurrence relations for modified Bessle's functions.
The goal of the course is to provide the student with an introduction to this piece of mathematical, including the fundamentals of solution techniques of a variety equations of (Bessel’s, laguerre, hypergeometric ) by using power series method. To achieve this goal,  students will, completion of homework assignments, and exams.

Student learning outcome:
1- To give the student the necessary information to deals with problems that could be a model for some mathematical problem.
2-To give the student the necessary mathematical tools for further study in applied  mathematics, where the Gamma function could be used in calculus.
3-To demonstrate the ability to use orthogonal functions (Legendre, Hermite, Bessle’s,..) in
approximating D.E, or expanding functions.
4-Will have the abilities like as Identification of differential equations which appear from special functions.
5- Will solve the equations to product the polynomials from the equations
6- Will apply the solving methods in the physical problems
7- To recognize special functions and learn its properties

The Topics:
Chapter One : Gamma and Beta functions: definition of Gamma and Beta functions,  convergence of Gamma and Beta functions, properties, examples, Dirichlet’s integral, some important results, Incomplete Beta and Gamma functions.
Chapter Two : Bessel’s function: solution of Bessel’s equation, parametric Bessel equation,  some particular forms, recurrence relations, generating function, Orthogonality, integral  representation, equation reducible to Bessel’s equation, spherical Bessel’s function.
Chapter Three :  Legendre polynomials: Legendre’s equation and its solution, rodrigue  formula for Legendre polynomials, Generating function, Recurrence relations, orthogonality  of Legendre polynomials, Expansion of functions.
Chapter Four :  Hermite function: Generate function, Hermite polynomials, orthogonality of Hermite polynomials, Properties, Recurrence relations.
Chapter five :  Laguerre Function: Laguerre differential equation, associated Laguerre polynomial, orthogonality of Laguerre polynomials, Properties, Recurrence relations.
Chapter Six : hebysheve polynomials: Chebysheve polynomials of the first and the second kind, Generating functions, recurrence relations, trigonometric form, orthogonality.
Chapter Seven : acobi polynomials: Basic formulas and algebraic properties, Generating  function and differential equation, Recurrence relations, orthogonality.
Chapter Eight :  Hypergeometric function: hypergeometric equation and its solution, Factorial function, Alternative form of hypergeometric function, Particular solution of hypergeometric equation, Properties, Gauss theorem, Vandermonde theorem.

Course Reading List and References:
- N. N. Lebedev,“Special Functions & Their Applications”. .
- Murray R. Spiegel, “Advanced Mathematics for Engineers and Scientists”. Schaum’s Outline Series. McGRaw-Hill. London, 1999.
- Roger C. McCann, ” Introduction to ordinary differential equations” . HBJ, INC. New york, 1982.
- G. Andrews , R. Askey & R. Roy , ”Special Functions”, Printed in united states, 2000.

Time series

Units : 4
Lecturer : Dr. Hussein Abdulrahman Hashem

Description : Course overview:
This course introduces the theory and practice of time series analysis, with an emphasis on practical skills. Having completed this course, you will be able to model and forecast a time series as well as read papers from the literature and start to do original research in time series analysis. More generally, you will acquire an appreciation for the role of dependence in statistical modelling.

Course objective:
To introduce a variety of statistical models for time series and cover the main methods for analysing these models. At the end of the course, the student should be able to
- Compute and interpret a sample spectrum
- Derive the properties of ARIMA and state-space models
- Choose an appropriate ARIMA model for a given set of data and fit the model using an appropriate package
- Compute forecasts for a variety of linear methods and models.

Student learning outcome:
On completion of the course the student should be able to
- present time series in an informative way, both graphically and with summary statistics,
- model time series to analyse the underlying structure(s) in both the time and frequency domains, and
- use the statistical software package ITSM to model and produce point and interval forecasts and interpret the results.

The Topics:
CHAPTER 1:Statistics Background for Forecasting
1.1 Numerical Description of Time Series Data
1.2 Use of Data Transformations and Adjustments
1.3General Approach to Time Series Modelling and Forecasting
CHAPTER 2: Regression Analysis and Forecasting
2.1 Least Squares Estimation in Linear Regression Models
2.2 Statistical Inference in Linear Regression
2.3 Prediction of New Observations
2.4 Model Adequacy Checking
2.5 Variable Selection Methods in Regression
2.6 Regression Models for General Time Series Data
CHAPTER 3: Exponential Smoothing Methods
3.1 First-Order Exponential Smoothing
3.2 Modelling Time Series Data
3.3 Second-Order Exponential Smoothing
3.4 Higher-Order Exponential Smoothing
3.5 Forecasting
3.6 Exponential Smoothing for Seasonal Data
3.7 Exponential Smoothers and ARIMA Models
CHAPTER 4: Autoregressive Integrated Moving Average (ARIMA) Models
4.1 Linear Models for Stationary Time Series
4.2 Finite Order Moving Average (MA) Processes
4.3 Finite Order Autoregressive Processes
4.4 Mixed Autoregressive-Moving Average (ARMA)Processes
4.5 Nonstationary Processes
4.6 Time Series Model Building
4.7 Forecasting ARIMA Processes
4.8 Seasonal Processes

Course Reading List and References: Books Recommended:
- Douglas C. Montgomery, Cheryl L. Jennings, and Murat Kulachi (2008). Introduction to
Time Series Analysis and Forecasting Wiley: New York.