Plemelj formulae

The first term on the right-hand side (RHS) of (D.2) gives 

Contrasting this with the Dirac 5-function definition, viz., 

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Some different forms of the Kramers-Kronig relations are presented in Table 22.1. Equations (22.55) and (22.56), called the Plemelj formulas, are obtained directly from consideration of causality in a linear system. Equations (22.57) and (22.58) are mathematically equivalent to equations (22.59) and (22.60), respectively, because... 

Let us consider the analytic continuations y(iz) = l//i(iz) and take the limit of this relationship as from above. The Plemelj Formulae (A2.2.9) give that... 

Note that it is singular at the end points of the cracks. This implies that the stresses will also be singular there. Consider now the expression for the derivative of the gap. From (4.2.5, 7) and the Plemelj formula (A2.2.10) ... 

The function 0(x) may be deduced from the other Plemelj formula ... 

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