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The main focus of this paper is to develop the solutions for the higher order difference equations with factorials and discrete exponential functions. Using these concept, we get the unique solution for the trigonometric exponential function for the initial valued problem. These results are verified using the numerical calculations.

Let Se∗ denote the class of analytic functions f in the open unit disk normalized by f(0)=f′(0)-1=0 and satisfying the condition zf′(z)/f(z)≺ez for |z|<1. The structural formula, inclusion relations, coefficient estimates, growth and distortion results, subordination theorems and various radii constants for functions in the class Se∗ are obtained. In addition, the sharp Se∗-radii for functions belonging to several interesting classes are also determined.

In this paper, we introduce a new class of analytic functions, denoted by S(ν,φ[sub.ϑ,e]), and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter et al. in relation to the exponential function. A significant contribution of this work is the derivation of sharp bounds for various coefficient-related problems within this class. The computational challenges involved in deriving these bounds were effectively addressed using Mathematica[sup.TM] codes. Additionally, figures illustrating the geometric properties and essential computations have been incorporated into the paper.

In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis.

In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \\mathbb{R}^{1+d} (d\\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

This paper focuses on the generalized Calogero–Bogoyavlenskii–Schiff equation to extract new complex solutions by using two analytical methods, namely, Bernoulli sub-equation function method, Modified exponential function method. For better understanding of physical meanings of solutions, simulations are reported by using a computational package program. Moreover, strain conditions for validity of complex solutions are also archived. Finally, a conclusion part completes the paper by mentioning the novelties of paper.

Background In the implementation of Digital Image Correlation (DIC), several post-processing methods have been developed to calculate reliable strain field. Nevertheless, achieving effective and easy-to-implement strain measurement for full-gradient strain fields continues to be a challenge. Objective The widely used pointwise least square (PLS) method is hard to get a balance between smoothing and accuracy when dealing with different deformation fields. A large strain calculation window may lead to over-smoothing, whereas a small strain calculation window may be insufficient to suppress noise. Methods A new exponential function and the exponential function weighted averaging (EFWA) method are proposed. The shape of the exponential function can be either sharp-topped or flat-topped, allowing the EFWA method to either preserve or smooth the original strain results. A straightforward and effective selection strategy for parameters of the exponential function is also provided, enabling the EFWA method to achieve self-adaptive post-processing. Results The calculation examples of synthetic images indicate that, the proposed EFWA method can consistently yield high measurement accuracy for unidirectional and multi-directional complex deformation fields and exhibits superior spatial resolution compared to the PLS method. The minimum Metrological Efficiency Indicator (MEI) value for the EFWA method is 1.72, compared to 4.67 for original results and 5.10 for the PLS method. The results of a tensile experiment carried out on an open-hole specimen indicate that, after the EFWA method is implemented, the strain results in areas away from the hole are effectively smoothed and the strain results in areas around the hole are preserved. Conclusions The proposed EFWA method can achieve effective and easy-to-implement strain measurement for full-gradient strain fields.

We establish the linear independence of values of the q-analogue of the exponential function and its derivatives at specified algebraic arguments, when q is a Pisot–Vijayaraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalised q-series.

In this article, a general fractional-order derivative of the Riemann-Liouville type with the non-singular kernel involving the Rabotnov fractional-exponential function is addressed for the first time. A new general fractional-order derivative model for the anomalous diffusion is discussed in detail. The general fractional-order derivative operator formula is as a novel and mathematical approach proposed to give the generalized presentation of the physical models in complex phenomena with power law. nema

In the present paper, we aimed to discuss certain coefficient-related problems for the inverse functions associated with a bounded turning functions class subordinated with the exponential function. We calculated the bounds of some initial coefficients, the Fekete–Szegö-type inequality, and the estimation of Hankel determinants of second and third order. All of these bounds were proven to be sharp.