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This brief monograph on the gamma function was designed by the author to fill what he perceived as a gap in the literature of mathematics, which often treated the gamma function in a manner he described as both sketchy and overly complicated. Author Emil Artin, one of the twentieth century's leading mathematicians, wrote in his Preface to this book, "I feel that this monograph will help to show that the gamma function can be thought of as one of the elementary functions, and that all of its basic properties can be established using elementary methods of the calculus." Generations of teachers and students have benefitted from Artin's masterly arguments and precise results. Suitable for advanced undergraduates and graduate students of mathematics, his treatment examines functions, the Euler integrals and the Gauss formula, large values of x and the multiplication formula, the connection with sin x, applications to definite integrals, and other subjects.
On a Class of Incomplete Gamma Functions with Applications by M. Aslam Chaudhry,Syed M. Zubair Pdf
The subject of special functions is rich and expanding continuously with the emergence of new problems encountered in engineering and applied science applications. The development of computational techniques and the rapid growth in computing power have increased the importance of the special functions and their formulae for analytic representations
This book is dedicated to the subject of the Gamma function and related topics. The Gamma Function is primarily intended for advanced undergraduates in science and mathematics. It is concise yet thorough and covers each of the most important aspects of the Gamma function. The Gamma function has important applications in probability theory, combinatorics and most, if not all, areas of physics. A large number of proofs and derivations of theorems and identities are covered in the book including: Analytic continuation of the factorials, properties via complex analysis, Holder's theorem, the Bohr-Mullerup theorem, the Beta function, Wallis's integrals, Wallis's product, product & reflection formulas, half-integer values, digamma and polygamma functions, series expansions, Euler-Mascheroni integrals, duplication & multiplication formulas, the Gamma and zeta function relationships, Hankel's contour integral representation, Stirling's formula, the Weierstrass factor theorem and the Mittag-Leffler theorem.
Author : C. Edward Sandifer Publisher : The Mathematical Association of America Page : 254 pages File Size : 48,8 Mb Release : 2014-11-19 Category : Mathematics ISBN : 9780883855843
Sandifer has been studying Euler for decades and is one of the world’s leading experts on his work. This volume is the second collection of Sandifer’s “How Euler Did It” columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician of history, this volume will leave you marveling at Euler’s clever inventiveness and Sandifer’s wonderful ability to explicate and put it all in context.
"This brief monograph on the gamma function by a major 20th century mathematician was designed to bridge a gap in the literature of mathematics between incomplete and over-complicated treatments. Topics include functions, the Euler integrals and the Gauss formula, large values of X and the multiplication formula, the connection with sin X applications to definite integrals, and other subjects. "--
Complex Variables with Applications by Saminathan Ponnusamy,Herb Silverman Pdf
Explores the interrelations between real and complex numbers by adopting both generalization and specialization methods to move between them, while simultaneously examining their analytic and geometric characteristics Engaging exposition with discussions, remarks, questions, and exercises to motivate understanding and critical thinking skills Encludes numerous examples and applications relevant to science and engineering students
Orthogonal Polynomials by Mama Foupouagnigni,Wolfram Koepf Pdf
This book presents contributions of international and local experts from the African Institute for Mathematical Sciences (AIMS-Cameroon) and also from other local universities in the domain of orthogonal polynomials and applications. The topics addressed range from univariate to multivariate orthogonal polynomials, from multiple orthogonal polynomials and random matrices to orthogonal polynomials and Painlevé equations. The contributions are based on lectures given at the AIMS-Volkswagen Stiftung Workshop on Introduction of Orthogonal Polynomials and Applications held on October 5–12, 2018 in Douala, Cameroon. This workshop, funded within the framework of the Volkswagen Foundation Initiative "Symposia and Summer Schools", was aimed globally at promoting capacity building in terms of research and training in orthogonal polynomials and applications, discussions and development of new ideas as well as development and enhancement of networking including south-south cooperation.
Mathematical Methods in Engineering by Nuno Miguel Fonseca Ferreira,José António Tenreiro Machado Pdf
This book presents a careful selection of the contributions presented at the Mathematical Methods in Engineering (MME10) International Symposium, held at the Polytechnic Institute of Coimbra- Engineering Institute of Coimbra (IPC/ISEC), Portugal, October 21-24, 2010. The volume discusses recent developments about theoretical and applied mathematics toward the solution of engineering problems, thus covering a wide range of topics, such as: Automatic Control, Autonomous Systems, Computer Science, Dynamical Systems and Control, Electronics, Finance and Economics, Fluid Mechanics and Heat Transfer, Fractional Mathematics, Fractional Transforms and Their Applications, Fuzzy Sets and Systems, Image and Signal Analysis, Image Processing, Mechanics, Mechatronics, Motor Control and Human Movement Analysis, Nonlinear Dynamics, Partial Differential Equations, Robotics, Acoustics, Vibration and Control, and Wavelets.
A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions by Jean-Luc Marichal,Naïm Zenaïdi Pdf
In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory.
Lectures on Probability Theory and Mathematical Statistics - 3rd Edition by Marco Taboga Pdf
The book is a collection of 80 short and self-contained lectures covering most of the topics that are usually taught in intermediate courses in probability theory and mathematical statistics. There are hundreds of examples, solved exercises and detailed derivations of important results. The step-by-step approach makes the book easy to understand and ideal for self-study. One of the main aims of the book is to be a time saver: it contains several results and proofs, especially on probability distributions, that are hard to find in standard references and are scattered here and there in more specialistic books. The topics covered by the book are as follows. PART 1 - MATHEMATICAL TOOLS: set theory, permutations, combinations, partitions, sequences and limits, review of differentiation and integration rules, the Gamma and Beta functions. PART 2 - FUNDAMENTALS OF PROBABILITY: events, probability, independence, conditional probability, Bayes' rule, random variables and random vectors, expected value, variance, covariance, correlation, covariance matrix, conditional distributions and conditional expectation, independent variables, indicator functions. PART 3 - ADDITIONAL TOPICS IN PROBABILITY THEORY: probabilistic inequalities, construction of probability distributions, transformations of probability distributions, moments and cross-moments, moment generating functions, characteristic functions. PART 4 - PROBABILITY DISTRIBUTIONS: Bernoulli, binomial, Poisson, uniform, exponential, normal, Chi-square, Gamma, Student's t, F, multinomial, multivariate normal, multivariate Student's t, Wishart. PART 5 - MORE DETAILS ABOUT THE NORMAL DISTRIBUTION: linear combinations, quadratic forms, partitions. PART 6 - ASYMPTOTIC THEORY: sequences of random vectors and random variables, pointwise convergence, almost sure convergence, convergence in probability, mean-square convergence, convergence in distribution, relations between modes of convergence, Laws of Large Numbers, Central Limit Theorems, Continuous Mapping Theorem, Slutsky's Theorem. PART 7 - FUNDAMENTALS OF STATISTICS: statistical inference, point estimation, set estimation, hypothesis testing, statistical inferences about the mean, statistical inferences about the variance.
Solved Problems in Analysis by Orin J. Farrell,Bertram Ross Pdf
Nearly 200 problems, each with a detailed, worked-out solution, deal with the properties and applications of the gamma and beta functions, Legendre polynomials, and Bessel functions. 1971 edition.
Mathematical Aspects of Computer and Information Sciences by Johannes Blömer,Ilias S. Kotsireas,Temur Kutsia,Dimitris E. Simos Pdf
This book constitutes the refereed proceedings of the 7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017, held in Vienna, Austria, in November 2017. The 28 revised papers and 8 short papers presented were carefully reviewed and selected from 67 submissions. The papers are organized in the following topical sections: foundation of algorithms in mathematics, engineering and scientific computation; combinatorics and codes in computer science; data modeling and analysis; and mathematical aspects of information security and cryptography.
This book is dedicated to the subject of the Gamma function and related topics. The Gamma Function is primarily intended for advanced undergraduates in science and mathematics. It is concise yet thorough and covers each of the most important aspects of the Gamma function. The Gamma function has important applications in probability theory, combinatorics and most, if not all, areas of physics. A large number of proofs and derivations of theorems and identities are covered in the book including: Analytic continuation of the factorials, properties via complex analysis, Holder's theorem, the Bohr-Mullerup theorem, the Beta function, Wallis's integrals, Wallis's product, product & reflection formulas, half-integer values, digamma and polygamma functions, series expansions, Euler-Mascheroni integrals, duplication & multiplication formulas, the Gamma and zeta function relationships, Hankel's contour integral representation, Stirling's formula, the Weierstrass factor theorem and the Mittag-Leffler theorem.