scholarly journals On Some Bounds for the Exponential Integral Function

2021 ◽  
Vol 4 (2) ◽  
pp. 28-34
Author(s):  
Kwara Nantomah
Keyword(s):  
Integral Function ◽  
Exponential Integral

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.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

A Limiting Approach for the Evaluation of Geometric Mean Transmittance in a Multidimensional Absorbing and Isotropically Scattering Medium

1984 ◽  
Vol 106 (2) ◽  
pp. 441-447 ◽  
Author(s):  
W. W. Yuen
Keyword(s):  
Geometric Mean ◽  
Accurate Estimate ◽  
Exact Expression ◽  
Integral Function ◽  
Scattering Medium ◽  
Exponential Integral ◽  
Physical Consideration ◽  
Upper Limits ◽  
Lower Limits ◽  
Transmittance Factor

The calculation of the geometric-mean transmittance factor between areas with an intervening absorbing and isotropically scattering medium is considered. While an exact expression for the factor is shown to be quite complicated, the upper and lower limits of the factor can be readily generated from physical consideration. Integral expressions for successively increasing (decreasing) values of the lower (upper) limits are obtained. For two-dimensional systems, these expressions are reduced to integrals involving Sn (x), a class of exponential integral function that has been tabulated in a previous work. Utilizing the kernel substitution technique, these integrals are evaluated analytically in closed form for some selected geometries. For cases with small optical thickness and large scattering albedo, both limits are shown to converge relatively slowly to the actual transmittance factor. But the decreasing difference between the two limits provides accurate estimate of the geometric-mean transmittance factor. Based on these results, some interesting conclusions concerning the effect of scattering on multidimensional radiative transmission are established.

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