This paper sheds more light on the co-rotational element formulation for beams with uniform cross-section. The co-rotational elements are commonly used in problems in which a structure undergoes a large deformation. In this study, the foregoing element obeys the Euler-Bernoulli beam assumptions. Unlike the formulations presented in the literature, in this paper, a number of local nodal coordina...

where all but one “masses” are equal, m1 = m −1, m2 = . . . = mn = 1, m is a real parameter, p̂j = i ∂ ∂xj , j = 1, . . . , n, and ℘ is the classical Weierstrass elliptic function. The case when m is integer is a special one: in that case a stronger version of integrability (the so-called algebraic integrability) was conjectured [2]. The first results in this direction were found in [3], where i...

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors stu...

As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K(m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9 to 19 obtained by truncating Taylor series exp...

The Coleman integral is a p-adic line integral. Double Coleman integrals on elliptic curves appear in Kim’s nonabelian Chabauty method, the first numerical examples of which were given by the author, Kedlaya, and Kim [3]. This paper describes the algorithms used to produce those examples, as well as techniques to compute higher iterated integrals on hyperelliptic curves, building on previous jo...

We developed the numerical procedures to evaluate the inverse functions of the complete elliptic integrals of the first and second kind, K(m) and E(m), with respect to the parameter m. The evaluation is executed by inverting eight sets of the truncated Taylor series expansions of the integrals in terms of m or of − log(1 −m). The developed procedures are (1) so precise that the maximum absolute...

In this paper, the perturbation method and Padé transformation are used to provide an approximate solution of elliptic integrals of the second kind and of complete integrals of the first kind. Besides, we used the obtained results to calculate an analytic expression for the period of a simple pendulum. The method has an acceptable accuracy for high values of the initial amplitude, compared to t...

Any product of real powers of Jacobian elliptic functions can be written in the form csm1 (u, k) ds2 (u, k) nsm3 (u, k). If all three m’s are even integers, the indefinite integral of this product with respect to u is a constant times a multivariate hypergeometric function R−a(b1, b2, b3; x, y, z) with halfodd-integral b’s and −a + b1 + b2 + b3 = 1, showing it to be an incomplete elliptic integ...

We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of applications.