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The Fourier Transform
Published in Patrick F. Dunn, Michael P. Davis, Measurement and Data Analysis for Engineering and Science, 2017
Patrick F. Dunn, Michael P. Davis
The sine is an odd function. Other properties of even and odd functions are as follows:The sum, difference, product, or quotient of two even functions is even.The sum or difference of two odd functions is odd.The product or quotient of two odd functions is even.The product or quotient of an even function and an odd function is odd.The sum or difference of an even function and an odd function is neither even nor odd, unless one of the functions is identically zero.A general function can be decomposed into a sum of even plus odd functions.
Orthogonal Expansions
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
Any function that is a linear combination of monomials xp with even (odd) powers p is an even (odd) function. Since the Maclaurin series for cosine function cosx=∑k⩾0(−1)kx2k(2k)! contains only even powers, it is an even function. Similarly, the sine function sinx=∑k⩾0(−1)kx2k+1(2k+1)! is an example of an odd function. A sum or difference of two or more even functions is an even function; for instance, x2+1−cosx is an even function. On the other hand, a product of two even or odd functions is an even function, while the product of an even and an odd function is an odd function. For instance, sin2x and cos2x are both even functions, while 2sinxcosx=sin(2x) is an odd function.
Scattering of SH-waves by oblique semi-elliptical notches in half space
Published in Waves in Random and Complex Media, 2022
Hui Qi, Fuqing Chu, Yang Zhang, Runjie Yang, Jing Guo
In the absence of a notch, the incident wave is reflected on the horizontal free surface (). And the incident wave and the reflected wave interfere to give the final wave field in the half-space. can be identified as Close to the notch, the incident wave and the reflected wave are scattered and diffracted by the elliptical-arc surface, then one more group of plane waves is generated. The waves are scattered and diffracted outgoing waves , which can be expressed as the series of elliptical wave functions. According to Sommerfeld’s radiation condition, the expression is as follows where and are the unknown coefficients to be determined. and are the even and odd functions of the third kind Mathieu function of order m, respectively. The total displacement field is the superposition of the incident wave, the reflected wave and the scattered wave i.e. . should satisfy the following boundary conditions:
Dynamic Stress Analysis of a Shallow Unlined Elliptical Tunnel under the Action of SH Waves
Published in Journal of Earthquake Engineering, 2022
Fuqing Chu, Hui Qi, Jing Guo, Guohui Wu
where . In addition, and are an even angular Mathieu function and an odd angular Mathieu function, respectively, with order m; and are the even and odd functions of the first modified Mathieu function of order m, respectively.
Student understanding of approximations in physics
Published in International Journal of Mathematical Education in Science and Technology, 2023
K. K. Mashood, Arvind Kumar, Anwesh Mazumdar
The mathematical background presumed in this study is binomial expansion, Taylor expansion of a function in single and multi-variable cases, including the important special cases of even and odd functions, and the basic notions of differential calculus. Several learning hurdles documented in this work can be traced to an insufficient understanding of the basic notions taught in their standard mathematics courses in calculus and analysis.