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Preparatory Tools of the Theory of Special Functions
Published in Willi Freeden, M. Zuhair Nashed, Lattice Point Identities and Shannon-Type Sampling, 2019
Willi Freeden, M. Zuhair Nashed
The function Pn(q;⋅):t↦Pn(q;t), t ∈ [ − 1, + 1], n = 0, 1, …, occuring in the addition theorem is called the Legendre polynomial of degree n and dimension q. polynomial!Legendre It is easily seen that Pn(q; · ) satisfies the following properties: Legendre polynomial!qD orthogonality
Nonlinear Propagation Equations and Solutions
Published in Shekhar Guha, Leonel P. Gonzalez, Laser Beam Propagation in Nonlinear Optical Media, 2017
Shekhar Guha, Leonel P. Gonzalez
The addition theorem for the sn function is given in Ref. [3] as () sn(u+v,γ2)=sn u⋅cn v⋅dn v+sn v⋅cn u⋅dn u1−γ2sn2u sn2v.
Mean second-order wave drift forces contour of a floating structure concept for wind energy exploitation
Published in C. Guedes Soares, Developments in Renewable Energies Offshore, 2020
Where h is the water depth, bq is the radius of the q cylinder, Im and Km are the m-th order modified Bessel functions of first and second kind, respectively. The first term in (8) represents the contribution of the incident wave field to the total wave potential around the q body. It consists of the undisturbed incident wave plus various orders of scattered waves emanating from the remaining bodies of array. These scattered wave fields can be expressed in the coordinate system of body q using a Bessel function addition theorem (Abramowitz & Stegun 1970).
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
Next, how to express and in elliptical coordinate system and make them to satisfy the stress-free condition of elliptical tunnel boundary is the focus and difficulty of this paper. Mathieu function addition theorem is the bridge of function transformation between different elliptic coordinate systems. Here are two elliptic cylindrical coordinate systems and as shown in Fig. 3. The region is defined as the region within the boundary of the circle with radius of and beyond the circle with radius of .
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
The methodology based on the Mathieu function is efficient for solution of elliptic boundary-value problems. With the maturity of numerical computation and computing power, it has been widely used in electromagnetism [19–22], acoustics [23], material science [24] and microwave technology [25]. Mathieu function addition theorem is a bridge of function transformation between different elliptic coordinate systems [26]. Its application makes the research of multi-envelope elliptic structure [21,27]and multi-elliptic structure [19,20,28]smooth, but few pieces of literature show that it is applied to the problem of elastic wave scattering. It can be seen that it is appropriate and meaningful to apply the Mathieu function and its related theories to the important problem of notching the elastic wave scattering. Thus, the main objective of this article is to determine the theoretical solution of the scattering by oblique semi-elliptical notches in an elastic half-space subjected to SH waves. Based on the research of Wong [13], this study obtained a more general method to apply more practical problems by adding Mathieu function addition theorem and multi-elliptical coordinate systems. The proposed wave system solution can be used as a benchmark for a new numerical method. In addition, the proposed method can be further applied to the strength design and dynamic analysis of intelligent mechanism materials by combining with the previous studies on piezoelectric and magnetized materials [8,9,29].
Optimal robust tracking by state feedback: infinite horizon
Published in International Journal of Control, 2022
Referring to the configuration of Figure 2, we take w = 0; this makes this configuration identical to the configuration of Figure 1. Recall Proposition 4.3, according to which the class of feedback functions is not empty; we restrict our attention to feedback functions . By Lemma 5.20, the response of the system is a weakly continuous function of the feedback function φ. In addition, Theorem 4.10(i) states that Σ is a weakly continuous function of its own input signal. As by (5.17), these facts imply that the response of is a weakly continuous function of the state-feedback function φ. Finally, since is when w = 0, our proof concludes.