We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant than the conventional one. We also discuss an interesting connection between our approach and the history of calculus. The article is written for readers who teach calculus and its applications. It might be accessible to students under a teacher's supervision and suitable for senior projects on calculus, real analysis, or history of mathematics.
Based on known definite integrals of Bessel functions of the first kind, we obtain exact solutions to unknown definite integrals using the method of integral transforms from Hankel's transform.
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations with one free parameter for these orthogonal polynomials. We also use orthogonality relations to determine corresponding definite integrals.
Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_pΨ_q(z)$ of one variable. Some consequences of this representation are explored by suitable choice of parameters. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced.
In this paper, we obtain analytical solutions of some definite integrals of Srinivasa Ramanujan [Mess. Math., XLIV, 75-86, 1915] in terms of Meijer's $G$-function by using Laplace transforms of $ \sin(βx^{2}),\cos(βx^{2}), x\sin(βx^{2})$ and $x\cos(βx^{2})$. Further, we obtain some infinite summation formulas connected with Meijer's G-function and numeric values of some infinite series
We give an algorithm to compute inhomogeneous differential equations for definite integrals with parameters. The algorithm is based on the integration algorithm for $D$-modules by Oaku. Main tool in the algorithm is the Gröbner basis method in the ring of differential operators.
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for theevaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.
We consider the expansion of an integral considered by F.G. Tricomi given by \[\int_{-\infty}^\infty x e^{-x^2}(\frac{1}{2}+\frac{1}{2}\mbox{erf}\,x)^{m} dx\] as $m\to\infty$. The procedure involves a suitable change of variable and the inversion of the complementary error function $\mbox{erfc}\,x$. Numerical results are presented to demonstrate the accuracy of the expansion. A second part examines an extension of an integral arising in airfoil theory.
We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, log Gamma(q) and log sin(q).
It is shown that the integrals of the Jacobi polynomials \begin{equation*}%\label{eq:Fn^J} \int_0^t (t-θ)^δP_n^{(α-\frac12,β-\frac12)}(\cos θ) \left(\sin \tfracθ2\right)^{2 α} \left(\cos \tfracθ2\right)^{2 β} dθ> 0 \end{equation*} for all $t \in (0,π]$ and $n \in \mathbb{N}$ if $δ\ge α+ 1$ for $α,β\in \mathbb{N}_0$ and $\max\{α,β\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \cite{BCX} that implies the strictly positive definiteness of the function $θ\mapsto (t - θ)_+^δ$ on the unit sphere $\mathbb{S}^{d-1}$ for $δ\ge \lceil \frac{d}{2}\rceil$ and the Polyà …
The Kampé de Fériet double series $F_{1:1;1}^{1:1;1}$ is studied through the solution to the associated first-order nonhomogeneous differential equation. It is shown that the integral of $t^{β+l}M(\cdot;β;λt)M(\cdot;β;-λt)$ over $t\in[0,T]$, $T\geq0$, $l=0,1,\ldots$, $\Reβ+l>-1$, is a linear combination of functions $F_{1:1;1}^{1:1;1}$. The integral is a generalization of a class of so-called Coulomb integrals involving regular Coulomb wave functions.
The definite integrals $ \int_{-1}^1x[P_ν(x)]^4\,\mathrm{d} x$ and $ \int_{0}^1x[P_ν(x)]^2\{[P_ν(x)]^2-[P_ν(-x)]^2\}\,\mathrm{d} x$ are evaluated in closed form, where $ P_ν$ stands for the Legendre function of degree $ ν\in\mathbb C$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.
As an application of Cauchy's Theorem we prove that $\int_0^1\arctan\left({\arctanh x-\arctan x\over π+\arctanh x-\arctan x}\right) {dx\over x}= {π\over 8}\log{π^2\over 8}$ answering a question first posted in Mathematics Stack Exchange and then in MathOverflow.
Deleting Items and Disturbing Mesh Theorems for Riemann Definite Integral and Their Applications
School of Mathematics and System Sciences, Beihang University（北京航空航天大学数学与系统科学学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results encompassing various aspects of the special function theory.
Let $\ell>0$ be arbitrary. We introduce the extremal quantities $$ G(\ell):=\frac{\sup_{f} \int_{-\ell}^{\ell} f\,dx}{\int_{-1}^1 f\,dx},\quad C(\ell):=\frac{\sup_{f} \sup_{a\in {\mathbb R}} \int_{a-\ell}^{a+\ell} f\,dx}{\int_{-1}^1 f\,dx}, $$ where the supremum is taken over all not identically zero non-negative positive definite functions. We are interested in the question: how large can the above extremal quantities be? This problem was originally posed by Yu. Shteinikov and S. Konyagin for the case $\ell=2$. In this note we obtain exact values for the right limits $G(k+0)$ and $C(k+0)$ at …
In this paper we investigate the following questions. Let $μ, ν$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying $$\int f dν\le C \int f dμ$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized, and what is their optimal value? We first discuss the problem in locally compact abelian groups. Then we make further specializations when the Borel measures $μ, ν$ are both either purely atomic or absolutely continuous with respect to a reference Haar measure. In addition, we prove…
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite integrals. Further, we derive some results as applications of these evaluations. Multi-variable cases of the derived results of this paper are also briefly discussed.
A new approximation of the discriminant of a second order periodic differential equation is presented as a recursive summation of the evaluation of its excitation function at different values of time. The new approximation is obtained, at first, by means of Walsh functions and then, by using some algebraic properties the dependence on the Walsh functions is eliminated. This new approximation is then used to calculate the boundaries of stability. We prove that by letting the summation elements number approach to infinite, the discriminant approximation can be rewritten as a summation of definit…
Among the several proofs known for $\sum_{n=1}^\infty{1/n^2} = {π^2/6}$, the one by Beukers, Calabi, and Kolk involves the evaluation of $\,\int_0^1 {\int_0^1{1/(1-x^2 y^2) \, dx} \, dy}$. It starts by showing that this double integral is equivalent to $\frac34 \sum_{n=1}^\infty{1/n^2}$, and then a non-trivial \emph{trigonometric} change of variables is applied which transforms that integral into $\,{\int \int}_T \: 1 \; du \, dv$, where $T$ is a triangular domain whose area is simply ${π^2/8}$. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it…
We present a systematic derivation of some definite integrals in the classical table of Gradshteyn and Ryzhik that can be reduced to the gamma function.
We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show in this paper the surprising fact that this integral does not depend on the parameters of the hypergeometric functions.
In this work we construct uniform asymptotic expansion of $\sn(t|m)$ - Jacobi when $m\to1-0$. The constructed expansion is valid over more than a half of period. The turning point is included into the interval of validity for the approximation. In addition we obtain the asymptotic formula for elliptic integral of the first kind and discuss the differences with the same formula from a handbook.
We present evalauations and provide proofs of definite integrals involving the function x^p cos^n x. These formulae are generalizations of 3.761.11 and 3.822.1, among others, in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik.
The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of posi…
We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms.
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.
In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's $M$ (or confluent hypergeometric ${}_1F_1$) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schlömilch- and Kapteyn-Bessel series recently established by Pogány, Süli, Baricz and Jankov Maširević.
We give three elementary proofs of a nice equality of definite integrals, which arises from the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences.
This paper presents an analytical closed-form solution to improper integral $μ(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is based on splitting the improper integral into an infinite sum of definite integrals with successive integer limits. The exact solution of every definite integral is obtained by making use of the binomial polynomial expansion, which then allows expression of the entire summation equivalently in terms of a weighted sum of Riemann zeta functions. It turns out that the solution fundamentally depends on whether or not $r$ is an integer. If $r$ is a n…
A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. The operational rules are described and the method is illustrated with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally suited for the evaluation of integrals associated with specific Feynman diagrams.
First some definite integrals of W. H. L. Russell, almost all with trigonometric function integrands, are derived, and many generalized. Then a list is given in Russell-style of generalizations of integral identities of Amdeberhan and Moll. We conclude with a brief and noncomprehensive description of directions for further investigation, including the significant generalization to elliptic functions.
The definite integrals $ \int_{-1}^1(1-x^2)^{(ν-1)/2}[P_ν(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(ν-1)/2}[P_ν(x)]^2P_ν(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^3\D x $ and $\int_{-1}^1x(1-x^2)^{(ν-1)/2}[P_{ν+1}(x)]^2P_{ν+1}(-x)\D x $ are evaluated in closed form, where $P_ν$ is the Legendre function of degree $ν$, and $ \Rν>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including parameters, then a holonomic difference-differential system for the integral can also be computed. In the algorithm, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.
We present here an approach to a computation of $ζ(2)$ by changing variables in the double integral using hyperbolic trig functions. We also apply this approach to present $ζ(n)$, when $n>2$, as a definite improper integral of single variable.
A modular relation of the form $F(α, w)=F(β, iw)$, where $i=\sqrt{-1}$ and $αβ=1$, is obtained. It involves the generalized digamma function $ψ_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $ζ_w(s, a)$. The limiting case $w\to0$ of this modular relation is a famous result of Ramanujan on page $220$ of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function $Ξ(t)$ as $α\to\infty$. Not only does it give the asymptotic estimate of the integral occurring in our modul…
