We present smooth approximations to $ \vert{x} \vert $ using sigmoid functions. In particular $ x\,erf(x/\mu) $ is proved to be better smooth approximation for $ \vert{x} \vert $ than $ x\,tanh(x/\mu) $ and $ \sqrt{x^2 + \mu} $ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.

This paper presents a new method for approximating the classical arcsine function. The proposed approximating methodology is simpler in its approach than other classical approaches and undeniably innovative. It is based on matrix representation besides the basic interpolation to approximate the inverse trigonometric function. It provides an efficient model which allows for reliable and precise calculations. The results are as per our knowledge unseen results in the previous literature.

In this review paper, we present a pellucid proof of how x tanh(x/µ) approximates |x|and is better than √x² +µ  when we are concernedwith accuracy.

In this article, we establish sharp trigonometric-polynomial bounds for unnormalized sinc function.

In this paper, we provide alternative proofs to some results proposed in the article "New inequalities involving circular, inverse circular, hyperbolic, inverse hyperbolic and exponential functions" authored by Yogesh J. Bagul.