On Some Bounds for the Exponential Integral Function

In 1934, Hopf established an elegant inequality bounding the exponential integral function. In 1959, Gautschi established an improvement of Hopf’s results. In 1969, Luke also established two inequalities with each improving Hopf’s results. In 1997, Alzer also established another improvement of Hopf’s results. In this paper, we provide two new proofs of Luke’s first inequality and as an application of this inequality, we provide a new proof and a generalization of Gautschi’s results. Furthermore, we establish some inequalities which are analogous to Luke’s second inequality and Alzer’s inequality. The techniques adopted in proving our results are simple and straightforward.

Calculation of Rail Impedance of Track Circuit Considering Earth Stratification

Rail impedance directly affects the transmission performance of track circuit . Considering the condition of earth stratification, for the difficult to calculate the rail impedance due to the semi-infinite integration interval and the oscillation of the integrand by using the Carson formula, The truncation method is proposed to divide the impedance formula is divided into definite integral and tail integral. The integral is approximated by the spline function, and the tail integral is calculated by using the exponential integral and Euler formula. Based on it, the rail impedance calculation formula of track circuit is obtained. The electromagnetic field model of track circuit with earth stratification is simulated by finite element method, and the correctness of the method is verified. Based on the formula, the influence of current frequency, soil depth and conductivity on rail impedance is studied. The relative error between the calculated results of rail impedance and the simulation results of finite element is within 5%. It can be seen that the formula has high accuracy and correctly reflects the law of rail impedance variation with current frequency, soil depth and resistivity. It provides a reliable reference for the theoretical calculation of rail impedance of track circuit.

A HARMONIC MEAN INEQUALITY CONCERNING THE GENERALIZED EXPONENTIAL INTEGRAL FUNCTION

In this paper, we prove that for $s\in(0,\infty)$, the harmonic mean of $E_k(s)$ and $E_k(1/s)$ is always less than or equal to $\Gamma(1-k,1)$. Where $E_k(s)$ is the generalized exponential integral function, $\Gamma(u,s)$ is the upper incomplete gamma function and $k\in \mathbb{N}$.

Binary response models comparison using the 𝛼-Chernoff divergence measure and exponential integral functions

In this paper, the families of binary response models are describing the data on a response variable having two possible outcomes and p p explanatory variables when the possible responses and their probabilities are functions of the explanatory variables. The α \alpha -Chernoff divergence measure and the Bhattacharyya divergence measure when α = 1 / 2 \alpha = 1/2 are the criterion functions used for quantifying the dissimilarity between probability distributions by expressing the divergence measures in terms of the exponential integral functions. The dependences of odds ratio and hazard function on the explanatory variables are also a part of the modeling.

On the p(x) approximation in the non-isothermal reaction kinetics by a generalized exponential integral the concept

A non-Arrhenius model based on the Mittag-Leffler function has been conceived as a basic concept. This approach allows modelling both sub-Arrhenius and super-Arrhenius behaviours and giving rise to modified temperature integrals.