The Mathematical Functions

2020

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Springer eBooks, 2016

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Lecture Notes in Computational Science and Engineering, 2015

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have honor to organize this meaningful conference. I believe that the purpose of this conference is not only sharing knowledge among the mathematician and scholars in related fields but also to hearten new generation of expertise in mathematics to realize the science and technology advancement. It is undeniable that there is mathematical harmony in science and technologies. Many disciplines like engineering, computer science, information technology, operational research, logistics management, risk management and many others are all the products of mathematics. Thus, it is essential that we must hold this annual conference as a stage for all scholars in finding new ideas and applications on Mathematics. Greatly thank to all supportive session including organizing committee, keynote speakers, invited speakers, paper reviewers, participants and sponsors. This event will not achieve without you all. Finally, I hope that the outcome of SEACMA 2013 will be pleasing and most useful to everybody.

The book, having eight chapters, offers one semester course (with four credit hours per week) on diverse topics usually included in the mathematics syllabus of the first year Engineering class of most Universities. The first chapter deals with the introductory topics in 2-dimensional coordinate geometry: straight lines and conic sections. A brief discussion of theory of equations is given in the second chapter. Vectors (mostly in 3-dimensional space) are introduced in the third chapter. Vector algebra and products of vectors are included. The next chapter deals with the matrices and determinants. Matrix algebra including multiplication of matrices is given. Further concepts such as rank of a matrix etc. are avoided. A comprehensive course on convergence of infinite series forms the subject mater of Chapter 5. Presuming that the students already had a first course on calculus only main concepts and results of differentiation and integration of functions of single variable have been given in the chapter 6. Power series, Maclaurin’s and Taylor’s expansions are explained in detail in the same chapter. Two Eulerian integrals, usually called Beta and Gamma functions, are also introduced and some properties of Gamma functions are given. Applications of vectors to geometry dealing with the vector equations of straight lines and planes are given in the Chapter 7. The last chapter deals with the partial derivation of functions of more than one variable. Both vector and scalar functions are considered and the vector differential operator of the first order is introduced. The total derivative of functions and famous Euler’s theorem on homogeneous functions are explained. Besides several worked out exercises (called Examples), Chapters 5, 7 and 8 are also supplemented by Problem-sets of unsolved exercises with necessary hints to the challenging ones. Tutorial sheets and Test Papers containing model questions and a short bibliography of the topics are provided. The alphabetical index added at the end makes the access to the contents faster. Chapters are divided into Sections, which are numbered chapter-wise. The discussion within the Sections is presented in the form of Definitions, Theorems, Corollaries, Notes and (solved) Examples. These subtitles within the Sections are numbered in decimal pattern. For instance, the equation number (c.s.e) refers to the eth equation in the sth section of Chapter c. When the number c coincides with the chapter at hand it is dropped. Adequate references to the previously appeared results are made in the text and unnecessary repetitions are avoided.

2020

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz 58 (2013) APPLICATIONS OF MATHEMATICS No. 4, 439-471

The series diverges when the limit of partial sums fails to exist, i.e., fails to approach a unique finite value. Exemple 1: Determine whether the following series converges or diverges: 00 * We will occasionally find it convenient to use the symbol E uk to denote E;~1 uk. lnfinite Series and lnfinlte Products 5 * This series in fact diverges, as can be shown by the integral test (Theorem 1.2). * The convergence of improper integrais is reviewed in Sec. 1.5. t For p > 1, the p series is also called the Riemann zeta functi.on (see Sec. 2.5.4). * We follow the convention that log x means log 10 x and ln x means log, x (ln x is known as the natural logarithm). lnfinite Series and lnfinite Products 9 1 O Chapter One Theorem 1.5 (Alternating-series test). If un >O, n = 1, 2, 3,... and limn__,."' Un =O, then the alternating series ~ (-l)llun converges (conditionally at least). Also the sum of a convergent alternating series always lies between the partial sums Sn and Sn+i for each n. Unlike in Theorem 1.1, we can conclude (conditional) convergence for an alternating series by showing that un ~O as n ~ oo. However, if we show that an alternating series converges by the alternatingseries test, we must further investigate the series to determine whether it also converges absolutely. This we do by applying another test to the related series of positive terms.

2014

In this paper, the author defines the Aleph function of two variables, which is a generalization of the I-function of two variables due to Sharma et al.[9]. In this regard the integral representation and applications of new function has been discussed. Similar results obtained by other authors follows as special cases of our findings.

2018

We have derived some new results for the Mellin transform formulas, as well as for the Gauss hypergeometric function. Also, we have found the connection between the Legendre functions of the second kind. Some of the results obtained we used in quantum mechanics of two charged particles of a continuous spectrum.

Beginning Julia Programming

2008

From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing complex systems can be conveniently written in terms of this generalization of the exponential function. The gamma function is then generalized and we generalize the factorial operation. Also a very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one-and two-tail stretched exponential functions. One obtains, as particular cases, the generalized error function, the Zipf-Mandelbrot probability density function (pdf), the generalized gaussian and Laplace pdf. One can also obtain analytically their cumulative functions and moments.

1987

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