IMPROPER INTEGRALS in .NET framework

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IMPROPER INTEGRALS
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De nition of an improper integral. Improper integrals of the rst kind (unbounded intervals). Convergence or divergence of improper integrals of the rst kind. Special improper integers of the rst kind. Convergence tests for improper integrals of the rst kind. Improper integrals of the second kind. Cauchy principal value. Special improper integrals of the second kind. Convergence tests for improper integrals of the second kind. Improper integrals of the third kind. Improper integrals containing a parameter, uniform convergence. Special tests for uniform convergence of integrals. Theorems on uniformly convergent integrals. Evaluation of de nite integrals. Laplace transforms. Linearity. Convergence. Application. Improper multiple integrals.
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FOURIER SERIES
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Periodic functions. Fourier series. Orthogonality conditions for the sine and cosine functions. Dirichlet conditions. Odd and even functions. Half range Fourier sine or cosine series. Parseval s identity. Di erentiation and integration of Fourier series. Complex notation for Fourier series. Boundary-value problems. Orthogonal functions.
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CONTENTS
Making Bar Code In Visual Studio .NET
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FOURIER INTEGRALS
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The Fourier integral. Equivalent forms of Fourier s integral theorem. Fourier transforms.
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GAMMA AND BETA FUNCTIONS
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The gamma function. Table of values and graph of the gamma function. The beta function. Dirichlet integrals.
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FUNCTIONS OF A COMPLEX VARIABLE
Functions. Limits and continuity. Derivatives. Cauchy-Riemann equations. Integrals. Cauchy s theorem. Cauchy s integral formulas. Taylor s series. Singular points. Poles. Laurent s series. Branches and branch points. Residues. Residue theorem. Evaluation of de nite integrals.
INDEX
Numbers
Mathematics has its own language with numbers as the alphabet. The language is given structure with the aid of connective symbols, rules of operation, and a rigorous mode of thought (logic). These concepts, which previously were explored in elementary mathematics courses such as geometry, algebra, and calculus, are reviewed in the following paragraphs.
SETS Fundamental in mathematics is the concept of a set, class, or collection of objects having speci ed characteristics. For example, we speak of the set of all university professors, the set of all letters A; B; C; D; . . . ; Z of the English alphabet, and so on. The individual objects of the set are called members or elements. Any part of a set is called a subset of the given set, e.g., A, B, C is a subset of A; B; C; D; . . . ; Z. The set consisting of no elements is called the empty set or null set.
REAL NUMBERS The following types of numbers are already familiar to the student: 1. Natural numbers 1; 2; 3; 4; . . . ; also called positive integers, are used in counting members of a set. The symbols varied with the times, e.g., the Romans used I, II, III, IV, . . . The sum a b and product a b or ab of any two natural numbers a and b is also a natural number. This is often expressed by saying that the set of natural numbers is closed under the operations of addition and multiplication, or satis es the closure property with respect to these operations. 2. Negative integers and zero denoted by 1; 2; 3; . . . and 0, respectively, arose to permit solutions of equations such as x b a, where a and b are any natural numbers. This leads to the operation of subtraction, or inverse of addition, and we write x a b. The set of positive and negative integers and zero is called the set of integers. 3. Rational numbers or fractions such as 2, 5, . . . arose to permit solutions of equations such as 3 4 bx a for all integers a and b, where b 6 0. This leads to the operation of division, or inverse of multiplication, and we write x a=b or a b where a is the numerator and b the denominator. The set of integers is a subset of the rational numbers, since integers correspond to rational numbers where b 1. p 4. Irrational numbers such as 2 and  are numbers which are not rational, i.e., they cannot be expressed as a=b (called the quotient of a and b), where a and b are integers and b 6 0. The set of rational and irrational numbers is called the set of real numbers. 1
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