Gudermannian function

Abstract: In the first part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical results given by Cauchy, Mellin, and Kronecker. In the second part, we give new formulas for some infinite series involving the hyperbolic cosine function.

Abstract: Hyperbolic imaginary unit and elliptic imaginary unit have corresponding complex spaces respectively. By utilizing Gudermannian function, the geometric relationships of two kinds of functions of a complex variable and complex planes can be established. The geometric connections of hyperbolic function and trigonometric function can be founded in the complex plane as well.

Abstract: Recently, a large number of integral formulas involving Bessel functions and their extensions have been investigated The objective of this paper is to establish four classes of integral formulas associated with the Struve functions, which are expressed in terms of the Fox-Wright function Among a variety of special cases of the main results, we present only six integral formulas involving trigonometric and hyperbolic functions

Abstract: For many years now, the problem of determining the arc of a hanging cable has been used in calculus texts as an application of hyperbolic functions. This example is certainly important as it indicates the use of hyperbolic functions in mechanics for engineering students. However, as seen in [1], there is another and perhaps simpler example of the application of hyperbolic functions. This example, based on towing a barge, also makes use of many more properties of the hyperbolic functions. Similar models, such as pulling a trailer, have appeared [6]. The problem can even be traced back to Leibniz, who posed it in terms of pulling a watch on a chain. Whatever the form of the problem, the path of the object has become known as a tractrix (see [2, p. 428]), and one could view the model as an early example in the theory of pursuit problems. In the next section we briefly review a few of the ways in which hyperbolic functions arise. The hanging cable problem is presented, but as a problem in the calculus of variations, rather than using the approach based on balancing forces. However, to show the idea of balancing forces, we develop the model of the hyperbolic arch. Then the barge problem is developed as an exercise in the use of the hyperbolic functions. In the process, we will see another natural and interesting way in which the Gudermannian function appears (see [4]).