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Basic Aspects of Fourier Series
Published in James S. Walker, Fast Fourier Transforms, 2017
It is easy to see that the product of two odd functions, or the product of two even functions, is an even function, while the product of an odd and an even function is an odd function. We leave as an exercise for the reader the proofs of the following facts: whenever g is odd on the interval [−L, L], then () ∫−LLg(x)dx=0
Hyperbolic functions
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
A graph of y=cosechx is shown in Fig. 12.4(a). The graph is symmetrical about the origin and is thus an odd function.
Fourier Trigonometric Series
Published in Russell L. Herman, An Introduction to Fourier Analysis, 2016
A similar computation could be done for odd functions. f (x) is an odd function if f (-x) = – f (x) for all x. The graphs of such functions are symmetric with respect to the origin, as shown in Figure 2.10. If one integrates an odd function over a symmetric interval, then one has that
Square-rebar corrosion-induced cover cracking and its time prediction for historical reinforced concrete buildings in China
Published in Journal of Asian Architecture and Building Engineering, 2022
Hui Jin, Qing Chun, Yiwei Hua, Shiqi Zhang
In Equations. (14, 15), are all constants that represent the rigid displacement. According to the symmetry, is an odd function of and is an even function of ; therefore, and . The boundary conditions for both bevel edges are fixed ends, and in the true sense, all points on the ends cannot move or rotate. Nevertheless, the fixed end is impossible in many realistic problems to obtain a polynomial solution. As a result, the boundary condition is simplified such that the middle point of the bevel edge is fixed, and cannot move or rotate in the horizontal direction (Fan, Gao, and Li 2006). Considering the displacement boundary condition for the left side and that , then:
The use of acoustic emission elastic waves as diagnosis method for insulated-gate bipolar transistor
Published in Journal of Marine Engineering & Technology, 2020
Artur Bejger, Maciej Kozak, Radosław Gordon
Therefore, the continuous Fourier transform to the analysis of Rxx(τ) can be applied. By transforming function x(t) defined in the time domain to the frequency domain ω, a complex function X(ω) can be obtained, i.e. continuous Fourier transform of x(t): and denoting this transform as Sx(ω) we get: Similarly, to the transform of the real signal x(t), this transform can be also represented on the plane of complex numbers and written in the following form: where Bearing in mind that the parity is a feature of the autocorrelation function and sin(ωτ) is an odd function, we get: and then:
Infinitely many solutions for a class of quasilinear Schrödinger equations involving sign-changing weight functions
Published in Applicable Analysis, 2019
For and , the Krasnoselskii genus of A is the least integer n such that there exists an odd function . The genus of A is denoted by . Set and if there exists no with the above property for any n.