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Examples and applications
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Exercise 4.3 (The Cauchy distribution). Consider the probability density function 1π11+x2 on the real line. Show that the mean and variance do not exist. Comment: One might want to say that the mean is 0 by symmetry, using the principal value integral. The definition of an improper integral requires letting the limits of integration tend to −∞ and +∞ separately. (Compare with Example 8.3 of Chapter 4 and its alternative derivation in Section 1 of Chapter 5.)
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Published in John Srdjan Petrovic, Advanced Calculus, 2020
We have seen in Section 6.7 that a very useful tool when trying to establish the convergence of an improper integral is the Comparison Test. The same is true in the multivariable situation. We will leave the proof as an exercise.
Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n
Published in International Journal of Mathematical Education in Science and Technology, 2023
Yet the topic could be quite challenging for elementary linear algebra students, the exploration of orthonormal Laguerre polynomials within an inner product space would still provide students with opportunities as enrichment activities to further strengthen their knowledge of orthonormality. Prior to the implementation of these activities, it seems crucial to highlight some prerequisite student pre-knowledge (linear algebraic, and/or digital tool use) might be needed. Such pre-knowledge includes without being limited to: (i) the understanding that an inner product results in a scalar; (ii) an understanding of elementary DGS-function graphing skills and integration syntaxes; (iii) an understanding of the meaning of definite integral and improper integral in the context of DGS; (iv) the understanding that visualizing an improper integral inner product involving Laguerre polynomials corresponds to a finite net area between the curve and the x-axis; (v) the understanding of the Gramian matrix as an orthogonality/orthonormality establishing/verification tool for a given set of vectors; (vi) proficiency in basic CAS-related syntax such as vectors, matrices, polynomials, integration.
Stack of non-uniformly loaded shear cracks in magnetoelectroelastic materials
Published in Mechanics of Advanced Materials and Structures, 2021
Thus, when new variables defined by are introduced, the integral equation (20) for can be expressed as with To conform with the Plemelj formulae, this improper integral must be construed as a Cauchy principal value integral. Its appropriate solution is readily deduced, using the basic classical results of for example Gakhov [29] and Muskhelishvili [30], to be with and
Fractional dynamical model to assess the efficacy of facemask to the community transmission of COVID-19
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Isa Abdullahi Baba, Musbahu Aminu Sani, Bashir Ahmad Nasidi
The improper integral (2.5) has the following properties