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Название работы: Tables of Integrals and Other Mathematical Data Автор: H.R. Dwight Загрузил: Denis aka Rock Lee Описание: The first study of any portion of mathematics should not be done from a synopsis of compact results, such as this collection. The references, although they are far from complete, will be helpful, it is hoped, in showing where the derivation of the results is given or where further similar results may be found. A list of numbered references is given at the end of the book. These are referred to in the text as \"Ref. 7, p. 32,\" etc., the page number being that of the publication to which reference is made.
Letters are considered to represent real quantities unless otherwise stated. Where the square root of a quantity is indicated, the positive value is to be taken, unless otherwise indicated. Two vertical lines enclosing a quantity represent the absolute or numerical value of that quantity, that is, the modulus of the quantity. The absolute value is a positive quantity. Thus, log | - 3| = log 3.
The constant of integration is to be understood after each integral. The integrals may usually be checked by differentiating.
In algebraic expressions, the symbol log represents natural or Napierian logarithms, that is, logarithms to the base e. When any other base is intended, it will be indicated in the usual manner. When an integral contains the logarithm of a certain quantity, integration should not be carried from a negative to a positive value of that quantity. If the quantity is negative, the logarithm of the absolute value of the quantity may be used, since log (— 1) = (2k + 1)хг will be part of the constant of integration (see 409.03). Accordingly, in many cases, the logarithm of an absolute value is shown, in giving an integral, so as to indicate that it applies to real values, both positive and negative.
Inverse trigonometric functions are to be understood as referring to the principal values.
Suggestions and criticisms as to the material of this book and as to errors that may be in it, will be welcomed.
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Название работы: The application of continued fractions Автор: Khovanskii A.N. Загрузил: Denis aka Rock Lee Описание: In modern mathematics the approximate representation of functions is ordinarily sought for in the form of a polynomial in the independent variable. In cases in which such polynomials are difficult to find, other numerical methods are used.
For this purpose approximations by rational functions of the independent variable have seldom been used. A characteristic of rational function approximations is that they may often successfully represent the given function in a domain of variation of the argument where the power series expansion of the function diverges and where, in consequence, in a great number of cases a polynomial approximation is inapplicable.
Furthermore, with the help of rational function approximations, the determination of the zeros and poles of the given function is greatly facilitated, since it is required to solve an algebraic equation of lower degree than that which occurs when using an approximation in the form of a polynomial.
Finally, the use of a rational function approximation tends to remove the necessity of computing high powers of the argument.
Thus the application of rational function approximations brings about a great simplification in many of the computing formulae.
That approximation by means of a rational function should have gained so small a currency is explained by the fact that the direct derivation of this function necessitates lengthy calculations. Furthermore the transition from one rational function approximation to another involves, in general, the recomputation of all coefficients contained in the numerators and denominators of these approximations. However methods exist allowing the derivation of arbitrarily many rational function approximations to the given function, in a manner not demanding complicated calculations. The most widely known methods of this type are based on the use of continued fractions. Подробнее по ссылке:http://www.tnu.in.ua/study/downloads...o=file&id=2985
Название работы: The Elements of Integration Автор: R.G. Bartle Загрузил: Denis aka Rock Lee Описание: There was a time when an undergraduate student of mathematics was expected to develop technique in solving problems that involved considerable computation; however, he was not expected to master theoretical subtleties such as uniform convergence or uniform continuity. The student was expected to be able to use the Implicit Function Theorem, but was not expected to know its hypotheses. The situation has changed. Now it is generally agreed that it is important for all students — whether future mathematicians, physicists, engineers, or economists — to grasp the basic theoretical nature of the subject. For, having done so, they will understand both the power and the limitation of the general theory and they will be better equipped to devise specific techniques to attack particular problems as they arise.
This text has developed from my experience in teaching courses in elementary real analysis at the University of Illinois since 1955. My audience has ranged from well-prepared freshman students to graduate students; the majority in these classes are usually not mathematics majors. Generally they have taken at least the equivalent of three semesters of non-rigorous calculus, including multiple integrals, vector calculus, line integrals, infinite series, and the like.
It would be desirable to have the students take a semester either in linear or modern algebra before this analysis course, for such a background facilitates the study of rigorous analysis. However, since the students I encounter do not all have this background, I purposely delay the study of analysis and first explore the notion of an ordered field to provide practice in giving proofs. Thus the first six sections of this text are mostly preparatory in nature; they can be covered in about three weeks in a normal class and more rapidly in a well-prepared one. Подробнее по ссылке:http://www.tnu.in.ua/study/downloads...o=file&id=2986
Название работы: The four-color theorem Автор: Fritsch R., Fritsch G. Загрузил: Denis aka Rock Lee Описание: During the university reform of the 1970s, the classical Faculty of Science of the venerable Ludwig-Maximilians-Universitat in Munich was divided into five smaller faculties. One was for mathematics, the others for physics, chemistry and pharmaceutics, biology, and the earth sciences. Nevertheless, in order to maintain an exchange of ideas between the various disciplines and so as not to permit the complete undermining of the original notion of \"universitas,\"1 the Carl-Friedrich-von-Siemens Foundation periodically invites the professors from the former Faculty of Science to a luncheon gathering. These are working luncheons during which recent developments in the various disciplines are presented by means of short talks. The motivation for such talks does not come, in the majority of cases, from the respective subject itself, but from another discipline that is loosely affiliated with it.
In this way, the controversy over the modern methods used in the proof of the Four-Color Theorem had also spread to disciplines outside of mathematics. I, as a trained algebraic topologist, was asked to comment on this. Naturally, 1 was acquainted with the Four-Color Подробнее по ссылке:http://www.tnu.in.ua/study/downloads...o=file&id=2987