Related papers: Some exponential and trigonometric integrals invol…
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are…
We consider several possible approaches to evaluating an integral involving the digamma function and a related logarithmic series.
This paper provides a Liouville principle for integration in terms of exponential integrals and incomplete gamma functions.
Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here.
We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic…
In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann…
A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as \[…
In the paper, the authors establish three kinds of double inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The…
This paper evaluates some generalised Euler sums involving the digamma function.
In this paper, we study the multiple integral $ \displaystyle I= \int_0^1 \int_0^1 \dots \int_0^1 f(x_1+x_2 + \dots +x_n) \, dx_1 \, dx_2 \, \dots \, dx_n$. A general formula of $I$ is presented. As an application, the integral $I$ with…
Criteria are given that kappa-deformed logarithmic and exponential functions should satisfy. With a pair of such functions one can associate another function, called the deduced logarithmic function. It is shown that generalized…
We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.
This paper is an enhanced version of a more than decade-older paper with a similar title. Many formulae involving both finite and infinite sums of digamma and polygamma functions up to quadratic order, few of which appear in standard…
This article revisits an integral of radical trigonometric functions. It presents several methods of integration where the integrand takes the form $\sqrt{1 \pm \sin x}$ or $\sqrt{1 \pm \cos x}$. The integral has applications in Calculus…
In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double…
In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the…
In this note we explore the relationship between the operation of convolution of functions and the Eulerian integrals. This approach allow us to obtain some expressions for the convolution of a certain class of functions in terms of the…
The Digamma and Polygamma functions are important tools in mathematical physics, not only for its many properties but also for the applications in statistical mechanics and stellar evolution. In many textbooks is found its develop almost by…
We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function.
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions;…