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Power Series Solutions and Special Functions
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
where each Pi(x) is a polynomial. The elementary functions consist of the algebraic functions; the elementary transcendental (or nonalgebraic) functions occurring in calculus—i.e., the trigonometric, inverse trigonometric, exponential, and logarithmic functions; and all others that can be constructed from these by adding, subtracting, multiplying, dividing, or forming a function of a function. Thus, y=tan[xe1/x+tan−1(1+x2)sinxcos2x−logx]1/3
Functions of Several Variables
Published in John Srdjan Petrovic, Advanced Calculus, 2020
We define elementary functions in the same way as in Section 3.7, i.e., these are constants, exponential and logarithmic functions, power functions, trigonometric functions and their inverses, as well as all functions obtained from the already listed through composition and combinations using the four arithmetic operations. The difference is that, having more than one independent variable, we need to replace the identity function by the functions of the form f (x1, x2, …, xn) = xi, 1 ≤ i ≤ n. For example, every polynomial in two variables is an elementary function. We leave the proof of this fact to the reader.
Analysis
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Not all integrals of elementary functions (sines, cosines, rational functions, and others) can be evaluated in terms of elementary functions. For example, the integral ∫e-x2dx $ \mathop \smallint \limits_{{}}^{{}} e^{{ - x^{2} }} dx $ is represented by the special function“erf(x)” (see page 475).
A novel solution to thermo-elasto-dynamic response for inhomogeneous orthotropic hollow cylinders
Published in Mechanics of Advanced Materials and Structures, 2023
Wen Lu, Chunxiao Zhan, Zhigen Wu
For an inhomogeneous cylinder, variation of material properties may be described as an elementary function (such as the power-law function [33] and the exponential function [24]) or piece-wise function with respect to the radial coordinate r. The elementary function is often used for FGM cylinders, and the piece-wise function can be applied for multilayered cylinders. In this section, by using the present method we provide the dynamic thermoelastic behavior of orthotropic hollow cylinders under the overall thermal shock and mechanical loading, and material properties (mass density ρ and elastic parameters cij) are given by where f (r) is an arbitrary function of the radial coordinate r, P0 denotes the reference value of material properties, e.g. ρ0 denotes the reference value of the mass density ρ, cij0 denotes the reference value of the elastic parameter component cij.
Integrability and linearizability of a family of three-dimensional quadratic systems
Published in Dynamical Systems, 2021
Waleed Aziz, Azad Amen, Chara Pantazi
(i) h = 0 and . First consider the case where the polynomial first integral does not depend on the variable x, so n = 0 and From relation (25) we have and the solution is with an arbitrary function in the variable Since the first integral is a polynomial and we have that must be zero and this means that , a contradiction. Now we consider that the first integral is of degree n>0 in the variable x and so The terms of in (25) satisfy The solution of the last linear partial differential equation is and is an arbitrary function in the variable Note that denotes the error function which is not an elementary function, see for example [34]. Since is a polynomial and , then from the last expression of it must be and therefore . This is a contradiction.
Design for additive manufacturing (DfAM) methodologies: a proposal to foster the design of microwave waveguide components
Published in Virtual and Physical Prototyping, 2019
Mathieu François, Frédéric Segonds, Mickaël Rivette, Simon Turpault, Patrice Peyre
By means of these representations, RF candidate parts – associated with an elementary function – are selected. This can be done by considering the dimensions of the part with the selected machines, specific desired characteristics in some regions incompatible with the AM technology, different materials for some parts, etc. If there are no obvious incompatibilities, the objective is to settle on all the elementary functions that compose the global function and run the third step.