
AU Mathematics III Sem III Unit 4 : Integral Transforms(Introduction, Definition, Fourier Integral, Sine and Cosine Integrals, Complex Forms of Fourier
Integral, Fourier Transform, Fourier and Cosine Transforms, Finite Fourier Sine and Cosine
Transforms, Properties of F  Transforms, Convolution Theorem for F  Transforms, Parseval s
Identity for Fourier Transforms, Fourier Transforms of the Derivatives of a Function,
Applications to Boundary Value Problems, Using Inverse Fourier Transforms only) 


1. The Integral of a Nonnegative function,Fatou's Lemma,Monotone Convergence Theorem(Mathematics) 
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In this unit, we introduce the concept of Lebesgue integral of nonnegative
measurable function , which is extended upto Fatou's lemma and Monotone
Convergence Theorem.
Title: Math 3.2 Measure Theory




2. Definition,Criterion for RiemannStieltjes Integral(Mathematics) 
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The RiemannStieltje's integral is the important generalization of the Riemann integral. The Riemann integral is a particular case of a more general integral.
Title: Math 2.2 Real Analysis II




3. The Properties and Classes of Integrable Functions(Mathematics) 
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In this unit we shall study the algebraic properties of the RiemannStieltje 's integral. i. e. sum, difference and product of two RS integrable functions are again RS integrable.
Title: Math 2.2 Real Analysis II




4. Differentiation of an Integral(Mathematics) 
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In this unit, we introduce the concept of the differentiation of an integral.
Title: Math 3.2 Measure Theory




5. Functions of Bounded Variations(Mathematics) 
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The concept of bounded variation helps in extending the theories of integration and differentiation. The definition of RiemannSteiltje's integral can be extended for the cases of monotonically non decreasing functions alpha on [a, b) by employing bounded variation notion. In this unit we shall study some properties of functions of bounded variation.
Title: Math 2.2 Real Analysis II




6. The Riemann Integral(Mathematics) 
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In this unit, we introduce the concept of Linear spaces. We give the
properties of Linear spaces and some theorems related to Linear spaces.
Finally, we explain the Linear operators and some theorems related to Linear
operators.
Title: Math 3.2 Measure Theory




7. Algebra of sets  Sigma Algebra and Borel Sets(Mathematics) 
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In this unit, we introduce the concept of algebra of sets. this unit also explains the concept of sigma algebra along with some examples. This unit provides an insight into Borel sets along with some examples.
Title: Math 3.2 Measure Theory




8. Metric Spaces, Completeness(Mathematics) 
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In this unit, we introduce the concept of metric spaces. We give the
properties of metric spaces and some theorems related to metric space.
Finally, we explain the Metric Completion Theorem.
Title: Math 3.3 Functional Analysis




9. Outer measure and measurable sets(Mathematics) 
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This unit explains Lebesgue outer measure and its properties. It also explains Lebesgue measurable sets and its properties (Theorems).
Title: Math 3.2 Measure Theory




10. Contents(Mathematics II) 
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This comprehensive and assimilable text introduces to the students the Mathematical concepts such as Matrices, Vector Calculus, Analytic Functions, Complex Integration and Laplace Transforms. The simple and systematic presentation will enable the students to grasp the concepts easily.
Title: Engineering Mathematics II
Published on: 09/02/19
Author:
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10








11. Setting up of First Order Differential Equations and their Solutions(Mathematics) 
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Differential equations occur quite frequently in our daily life. The motion of an object can always be associated with a differential equation. The change in prices of commodities, the flow of fluids, the concentration of chemicals etc., often lead to differential equations. Such equations may depend on one or more independent variables. Further, it may include the derivatives of the first or high
......
Title: Math 3.4 Mathematical Modeling




12. The Metric Topology(Mathematics) 
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Our main objective is to study the definition and some properties of metric topology in this unit.
Title: Math 3.1 Topology




13. Mathematical Modeling of Some Fundamental Problems(Mathematics) 
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Mathematical Modeling in terms of differential equations arises when the situation modeled involves some continuous variables varying with respect to other continuous variables and we have reasonable hypothesis about the rate of change of dependent variables with respect to independent variables. Mathematical models in terms of ordinary differential equations will be studied in this unit and the u
......
Title: Math 3.4 Mathematical Modeling




14. Continuous Functions(Mathematics) 
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We define continuous function and homeomorphism in this unit.
Title: Math 3.1 Topology




15. Measurable functions, Littlewood's three principles(Mathematics) 
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This unit defines measurable functions along with examples. It also explains the Littlewood's three principles and states some theorems on measurable functions.
Title: Math 3.2 Measure Theory




16. Banach Contraction Mapping Theorem and Applications(Mathematics) 
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In this unit, we introduce the concept of Banach Contraction Mapping
Theorem and its Applications.
Title: Math 3.3 Functional Analysis




17. Closed Sets, limit points, Hausdorff spaces(Mathematics) 
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This unit deals with the definition of closed sets, limit points and neighborhood in a topological space. The Hausdorff spaces will we defined in the unit which deals with separation axioms.
Title: Math 3.1 Topology




18. Programming Languages and Operating Systems(Mathematics) 
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System software is a program or a group of programs written for computer system management. The system software's are developed by the manufactures. They are supervisory programs that help executing the user's program effectively. The operating systems, language processors (such as interpreters and compliers), and utility programs (such as loaders and linkers) are examples of system software are s
......
Title: Math 3.5 Computer Programming




19. Building Blocks of Computer Programs(Mathematics) 
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The specification of the sequence of computational steps in a particular programming language is termed as a program. The task of developing programs is called programming. The person engaged in programming activity is called programmer. In this unit, we study on program analysis, Algorithm development, Flow chart, Decision tables and pseudo code.
Title: Math 3.5 Computer Programming




20. Parallel Computers and Network(Mathematics) 
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Parallel computing is a form of computation in which many instructions are carried out simultaneously operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently (in parallel). A network computing environment is one in which an organization has linked together personal computers that have been cOlU1ected into a network.
Title: Math 3.5 Computer Programming




21. Lebesgue measure, nonmeasurable sets(Mathematics) 
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This unit defines Lebesgue measure and proves certain theorems given in this unit. It also helps in distinguishing measurable and non measurable sets.
Title: Math 3.2 Measure Theory




22. Fundamental Aspect of Computers(Mathematics) 
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This help file presumes that you have little or no experience with the device commonly known as the PC (personal computer). Computers have become the basic necessity of any organization with serious objectives, They have made great inroads into everyone's day to day life and thinking. They are used for all sorts of problem solving ranging from simple addition to highly complex calculations in the
......
Title: Math 3.5 Computer Programming




23. Introduction to mathematical Modeling(Mathematics) 
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A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences such as physics, biology, earth science, meteorology and engineering disciplines like computer science, artificial intelligence, but also in the social sciences Physi
......
Title: Math 3.4 Mathematical Modeling




24. Cauchy's Integral Formula and its Consequences(Mathematics) 
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In this unit, we establish another fundamental result known as Cauchy's integral formula using Cauchy's theorem. As application of Cauchy's integral formula. we also deduce Morera's theorem, Liouville's theorem and Fundamental theorem of Algebra.
Title: Math 1.3 Complex Analysis  I




25. Limitations of Mathematical Modeling(Mathematics) 
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A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. In this regard we have discussed the limitations of modeling in following sections.
Title: Math 3.4 Mathematical Modeling






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