一类含指数函数的不定积分的探讨

通常的微积分教材中,可以计算的不定积分主要包括有理函数、三角函数有理式和某些含根式的函数.受几个具体函数积分方法的启发,证明了一类含有指数函数的积分公式,该结果包含了微积分教材中用常规积分方法较难推演的一类含指数函数的积分.     

贝塞尔函数在计算几种特殊三角函数积分中的应用

<正>在工程上特别是科技仪器理论中有时会遇到一些被积函数中三角函数又含有三角函数的在工程上特别是科技仪器理论中有时会遇到一些被积函数中三角函数又含有三角函数的份殊二用幽数附积分,.而这些积分一般是不能直接用求何应用贝塞尔函数来计算下面两种特殊的三角函数积分特殊三角函数的积分,而这些积分;能     

一种含三角函数式积分的特殊方法

与三角函数有关的不定积分是一类常见的重要积分 ,由于三角函数有许多特殊性质 ,如 :各三角函数之间有三角公式相联系着、三角函数的导数仍然是三角函数等 ,使得一些三角函数的积分方法非常灵活 ,因此技巧性也较强 .常规的教学中一般介绍凑微分法、换元积分法、分部积分法、三角函数有理式积分法等 ,对于有些被积函数较复杂的的积分用上述方法求可能较繁琐 .本文介绍一种计算三角函数式积分的特殊方法——“相关积分法”,这种方法的步骤是根据不定积分 I的被积函数 ,作出相关辅助不定积分 I1,I2 ,… ,利用 I和 I1,I2 ,…的不同线性组合 ,…     

一类二重积分的解法剖析及其推广

本文利用三角函数的性质分析并求解了一类二重积分,根据此二重积分中被积函数的对称性,将其推广到一类三重积分,得到了其积分值.     

关于一类大区间积分的化简问题

<正> 近年来微积分学在其理论、计算方法和应用等方面又有许多的新成果.但对于大量“积不出来”的一类定积分的研究仍是一个问题.本文得到的若干个新的公式,可将一类含有三角函数的在大区间[o,uπ]上的积分化为小区间[o,π/2]上的积分.从而可避开原函数而简便地求出其值,或可简化定性分析与近似计算.     

对数三角函数的定积分

定义了三种积分表示的两元函数.这些两元函数有伽马函数表示,可以展开为幂级数.在积分符号内展开被积函数,先积分,再求和,也得到级数展开.对比展开系数,就得到一些对数三角函数定积分的值.选取合适的围道,得到其他两类对数三角函数定积分的值.     

反三角函数相关的一些积分计算

在高等数学教学中,对于反三角函数相关的积分计算涉及的不多,一般用分部积分计算,本文在常规求解方法的基础上,给出了几类反三角函数相关的积分计算通项形式.     

三角函数的定积分

利用围道积分和参数展开,得到了一类含三角函数定积分的值.     

用凑微分法解三角函数有理式的积分

在不定积分的计算中,凑微分法是一种极为重要的方法.它的运用范围广泛,而且计算量较小,许多类型函数的积分都可以优先考虑应用这种方法.三角函数有理式的积分,用凑微分法通常是有效而较为简便的.     

一类三角函数有理式积分的简便求法

一类三角函数有理式积分的简便求法段玉珍（安徽电力职工大学）三角函数有理式的积分，从理论上说，它总可以通过万能代换化为有理函数的积分，但是有些类型的三角函数有理式的积分，用万能代换化为有理函数的积分往往比较繁，有的因形式过于复杂而行不通。由于这个缘故，...     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

Sharp inequalities for trigonometric polynomials with respect to integral functionals

The problem on sharp inequalities for linear operators on the set of trigonometric polynomials with respect to integral functionals \(\int_0^{2\pi } {\phi \left( {\left| {f\left( x \right)} \right|} \right)dx}\) is discussed. A solution of the problem on trigonometric polynomials with given leading harmonic of least deviation from zero with respect to such functionals over the set of all functions φ defined, nonnegative, and nondecreasing on the semiaxis [0,+∞) is given.     

Loaded differential operators: Convergence of spectral expansions

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.     

Approximation of classes of analytic functions by a linear method of special form

We establish asymptotic equalities for the least upper bounds of deviations of trigonometric polynomials generated by a linear approximation method of a special form on classes of convolutions of analytic functions in the uniform and integral metrics.     

The calculus of the trigonometric functions

The trigonometric functions entered “analysis” when Isaac Newton derived the power series for the sine in his De Analysi of 1669. On the other hand, no textbook until 1748 dealt with the calculus of these functions. That is, in none of the dozen or so calculus texts written in England and the continent during the first half of the 18th century was there a treatment of the derivative and integral of the sine or cosine or any discussion of the periodicity or addition properties of these functions. This contrasts sharply with what occurred in the case of the exponential and logarithmic functions. We attempt here to explain why the trigonometric functions did not enter calculus until about 1739. In that year, however, Leonhard Euler invented this calculus. He was led to this invention by the need for the trigonometric functions as solutions of linear differential equations. In addition, his discovery of a general method for solving linear differential equations with constant coefficients was influenced by his knowledge that these functions must provide part of that solution.     

Rational interpolation and approximate solution of integral equations

In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equation.     

Best approximations of classes of functions defined by means of the modulus of continuity

This paper is devoted to an exact solution of problems of best approximation in the uniform and integral metrics of classes of periodic functions representable as a convolution of a kernel not increasing the oscillation with functions having a given convex upwards majorant of the modulus of continuity. The approximating sets are taken to be the trigonometric polynomials in the case of the uniform and integral metrics, and convolutions of the kernel defining the class with polynomial splines in the case of the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 579–589, May, 1992.     

Inversion of a logarithmic operator defined on a regular set of arcs lying on a circle

The theory of singular integral equations is used to derive simple inversion formulas for a logarithmic operator defined on a contour consisting of an arbitrary number of identical arcs lying on a circle at an equal angular spacing. The action of the inverse operator on trigonometric functions is calculated, and the moments of the inverse operator with trigonometric functions are found. Even simpler formulas are derived in the approximation of small arcs.     

On exact estimates of the convergence rate of fourier series for functions of one variable in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript>[–π, π]

The work is devoted to exact estimates of the convergence rate of Fourier series in the trigonometric system in the space of square summable 2π-periodic functions with the Euclidean norm on certain classes of functions characterized by the generalized modulus of continuity. Some N-widths of these classes are calculated, and the residual term of one quadrature formula over equally spaced nodes for a definite integral connected with the issues under consideration is found.     

Estimates of the local convergence rate of spectral expansions for even-order differential operators

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a wide class of even-order ordinary differential operators defined on a finite interval. These expansions are compared with the trigonometric Fourier series expansions of the same functions in the integral or uniform metric on an arbitrary interior compact set of the main interval as well as on the entire interval. We show the dependence of the equiconvergence rate of these expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the existence of infinitely many associated functions in the system of root functions.