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We generalize the exponential function to model instantaneous relative growth. The modified function is defined by a linear relationship between a continuous quantity (rather than time) and logarithmic relative growth. The corresponding formula is ln[q'(t)/q(t)] = a+b·q(t) where q'(t)/q(t) is instantaneous relative growth of a quantity q, t refers to time, a denotes initial logarithmic relative growth, and b is a shape parameter in terms of its sign, as well as a scaling parameter in terms of its magnitude. For calculating q(t), the exponential integral Ei[-b·q] = ∫(exp[-b·q]/q)dq is needed. The problem of taking the inverse of Ei[x] = zEi is addressed. In order to distinguish two possible solutions for given zEi, we define the two inverse functions Ei(-1)x > 0[zEi] and Ei(-1)x < 0[zEi]. An indirect method for their numerical evaluation is developed. With the generalized exponential function, one can model sigmoid growth (b < 0), exponential growth (b = 0), and explosive growth (b > 0), where the term \"explosive growth\" refers to a relative growth rate that increases with time. The resulting formula of generalized exponential growth is where qC is a calibrating quantity at time tC. For b = 0, the two functions equal the (standard) exponential function q(t) = qC· exp[(t-tc)·exp[a]]. In the case of sigmoid growth, the inflection point quantity is -1/b, which depends only on one parameter (b). Negative growth can be modeled by substituting t-tC with tC - t. Any two points of logarithmic relative growth can be connected unambiguously with the generalized exponential function, to derive the corresponding function of q(t). Furthermore, we derive formulas for the conversion of a segmented curve of logarithmic relative growth as a function of time, into an equivalent growth curve of q(t).Finally, the generalized exponential function is compared with a 2nd-degree polynomial and the nonlinear Schnute function. In conclusion, the generalized exponential function is useful for modeling a path of changing relative growth continuously, and to translate it into a growth curve of quantity as a function of time.

PurposeIn this paper, the author introduces a degenerate exponential integral function and further studies some of its analytical properties. The new function is a generalization of the classical exponential integral function and the properties established are analogous to those satisfied by the classical function.Design/methodology/approachThe methods adopted in establishing the results are theoretical in nature.FindingsA degenerate exponential integral function which is a generalization of the classical exponential integral function has been introduced and its properties investigated. Upon taking some limits, the established results reduce to results involving the classical exponential integral function.Originality/valueThe results obtained in this paper are new and have the potential of inspiring further research on the subject.

In the literature, there exists a closed form solution to the remaining life expectancy at age x when mortality is governed by the Gompertz law. This expression contains a special function that allows us to construct high-accuracy approximations, which are also helpful in assessing the elasticity of life expectancy with respect to the model parameters. However, to my knowledge, a similar formulation for life disparity does not exist, and as a consequence, it does not exist for life table entropy either.

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura–Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.

Abstract In this paper, we introduce a q , ωq,ω -analog of Tricomi expansion based on Hahn’s difference operator. Some properties of q , ωq,ω -Tricomi expansion are derived and proved in terms of incomplete q , ωq,ω -gamma functions. Also, a q , ωq,ω -analog of the exponential integral is presented as a series expansion of incomplete q , ωq,ω -gamma functions and shown to be a limiting case of a q , ωq,ω -Tricomi expansion.

The practical use of computational thermo-fluid dynamics (CFD) for food thermal process calculations still appears very premature due to both the high costs and the inhomogeneity and anisotropy of foods. Therefore, the traditional formula method with both Ball and Stumbo’s tables is still widely used due to its accuracy and safety. In both cases, the calculations require consulting and interpolating data from the respective tables, making the procedure slow and prone to human errors. The computerization of Ball’s tables to speed up and automate the calculations with a new mathematical approach based on the substitution of the integral exponential function and the initial cooling hyperbola has already been developed. The high accuracy obtained, superior to the direct regression of the table data, suggested adopting it also in the computerization of Stumbo’s tables. However, the latter are 14 times larger than those of Ball due to the extension of the thermo-bacteriological parameter z up to over 100 °C and the variability of the cooling lag factor Jcc. Therefore, the mathematical modelling was modified using an additional function, dependent on z and Jcc. The results obtained with the mathematical modelling showed a mean relative error and the standard deviation with respect to the Stumbo’s tables equal to MRE ± SD = 0.62% ± 1.29%. Further validation was obtained by calculating the thermal process time for different lethalities and thermo-bacteriological parameters with MRE ± SD compared to the Stumbo tables equal to 1.04% ± 0.82%.

In this work, we are interested in studying bending, buckling stability, and dynamic behaviors of functionally graded nanobeams reinforced by carbon nanotubes (FG-CNTRC) by a novel shear deformation theory. This model is simple and efficient based on the new polynomial-exponential integral shear deformation theory including the effect of size taking into account the effects of the Winkler–Pasternak elastic foundations. The polynomial-exponential transverse shear function is incorporated to better represent a new displacement field that includes indeterminate integral terms. It is presumed that the material possessions of FG-CNTRC are diverse along the thickness direction employing distinct four distributions of carbon nanotubes (NT-CNTs). The nonlocal elasticity offered by Eringen is used to involve size effects in this approach. The impacts of numerous factors such as the volume fraction of CNTs, the nonlocality, and the impacts of the elastic foundation on the response of the FG-CNTRC beams are examined.

Ramanujan’s approximation to the exponential function is reexamined with the help of Perron’s saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion coefficients may be given in closed form. Ramanujan’s approximation to the exponential integral is treated similarly.

A semi-analytical solution for the time dependence of the concentration of the intermediate is derived, in the case of two parallel mixed second order reactions followed by a first order reaction. The solution is restricted to equal initial concentrations for the reactants, and it is connected to the exponential integral. From the solution and through a proper choice of the dimensionless time (u) and concentration of the intermediate (y) one obtains a very simple relation between the maximum concentration of the intermediate (y max ) and the time associated with this concentration (u max ). This relation is governed by a parameter (β) which depends on the three rate constants and on the initial concentration. The smaller is β the larger are u max and y max , and the slower is the decay of y. An approximate expression connecting u max and β, has also been obtained, and it yields maximum errors of ~ 8% and ~ 15% for u max and y max , respectively. The obtained expression can be very useful from the experimental point of view, as it allows an a priori selection of the most suitable experimental technique to detect the intermediate, simply comparing its time resolution with t max (that is, u max transformed to a time unit). An illustrative calculation is also discussed.

A practical and important problem encountered during the atmospheric re-entry phase is to determine analytical solutions for the space vehicle dynamical equations of motion. The author proposes new solutions for the equations of trajectory and flight-path angle of the space vehicle during the re-entry phase in Earth’s atmosphere. Explicit analytical solutions for the aerodynamic equations of motion can be effectively applied to investigate and control the rocket flight characteristics. Setting the initial conditions for the speed, re-entering flight-path angle, altitude, atmosphere density, lift and drag coefficients, the nonlinear differential equations of motion are linearized by a proper choice of the re-entry range angles. After integration, the solutions are expressed with the Exponential Integral, and Generalized Exponential Integral functions. Theoretical frameworks for proposed solutions as well as, several numerical examples, are presented.