scholarly journals Definite Integrals using Orthogonality and Integral Transforms

2012 ◽  
Author(s):  
Howard S. Cohl
Keyword(s):  
Integral Transforms ◽  
Definite Integrals

Disturbance produced in an elastic half-space by impulsive normal pressure

1964 ◽  
Vol 60 (3) ◽  
pp. 683-696 ◽  
Author(s):  
M. Mitra
Keyword(s):  
Half Space ◽  
Surface Displacement ◽  
Normal Pressure ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Point Load ◽  
Definite Integrals ◽  
Operational Solution ◽  
Method Of Solution ◽  
Pressure Area

1. Introduction. The disturbance produced in an elastic half-space by different types of buried sources have been studied by various authors. However, the only exact solutions of problems for the half-space in which the disturbance is produced by impulsive surface tractions are due to Pekeris (9) and Chao (4), though for the half-plane such problems have been studied by de Hoop (5), Ang (1) and Mitra (8). Pekeris (9) evaluated the surface displacement produced by a normal point load on the surface using a method of inversion of the operational solution developed earlier (Bateman & Pekeris (2)). A finite pressure-area, however, represents physical situations better than a point load. In this paper, the displacement produced in the half-space by uniform impulsive pressure acting over a circular portion of the surface has been obtained in terms of definite integrals. On the surface, the displacement has been split up into a number of terms some of which represent the P-wave contribution, while others give the Rayleigh-wave contribution to the displacement. The method of solution involves the use of integral transforms and Cagniard's (3) method; Pekeris's method is not applicable in this case since the appropriate transformation of an integral along the real axis to one along the imaginary axis cannot be carried out.

scholarly journals The Mellin Transform of Logarithmic and Rational Quotient Function in terms of the Lerch Function

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Current Literature ◽  
Mellin Transform ◽  
Integral Transforms ◽  
Complex Numbers ◽  
Definite Integrals ◽  
Famous Book ◽  
General Complex

Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by ∫ 0 ∞ x m − 1 log k a x / β 2 + x 2 γ + x d x , where the parameters a , k , β , and γ are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.

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