Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π

A Catalan constant inspired integral odyssey

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by $${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$ Scott in [1] quipped that this constant seemed almost as useful as the more widely known Euler–Mascheroni constant γ, particularly in the evaluation of definite integrals. And like γ, Catalan's constant continues to remain one of the most inscrutable constants in mathematics where the question concerning its irrationality is not settled.

Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series

We present a method using contour integration to evaluate the definite integral of arctangent reciprocal logarithmic integrals in terms of infinite sums. In a similar manner, we evaluate the definite integral involving the polylogarithmic function L i k ( y ) in terms of special functions. In various cases, these generalizations give the value of known mathematical constants such as Catalan’s constant G, Aprey’s constant ζ ( 3 ) , the Glaisher–Kinkelin constant A, l o g ( 2 ) , and π .

Euler Sums and Integral Connections

In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric and hyperbolic type functions to generate many new Euler sum identities. We also give some new identities for Catalan’s constant, Apery’s constant and a fast converging identity for the famous ζ ( 2 ) constant.

Dual Taylor Series, Spline Based Function and Integral Approximation and Applications

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series approximation. It is shown that a spline based series approximation to an integral yields, in general, a higher accuracy for a set order of approximation than a dual Taylor series, a Taylor series and an antiderivative series. A spline based series for an integral has many applications and indicative examples are detailed. These include a series for the exponential function, which coincides with a Padé series, new series for the logarithm function as well as new series for integral defined functions such as the Fresnel Sine integral function. It is shown that these series are more accurate and have larger regions of convergence than corresponding Taylor series. The spline based series for an integral can be used to define algorithms for highly accurate approximations for the logarithm function, the exponential function, rational numbers to a fractional power and the inverse sine, inverse cosine and inverse tangent functions. These algorithms are used to establish highly accurate approximations for π and Catalan’s constant. The use of sub-intervals allows the region of convergence for an integral approximation to be extended.

Integrals evaluated in terms of Catalan's constant

Catalan's constant, named after E. C. Catalan (1814-1894) and usually denoted by G, is defined byIt is, of course, a close relative ofThe numerical value is G ≈ 0.9159656. It is not known whether G is irrational: this remains a stubbornly unsolved problem. The best hope for a solution might appear to be the method of Beukers [1] to prove the irrationality of ζ (2) directly from the series, but it is not clear how to adapt this method to G.