Complex Variable Integral Equations

The Kolosov-Muskhelishvili formulae (25.1), (25.2) show that two functions φ(z) and Ψ(z), holomorphic in a simply connected region, define the complex displacements and the resultant force. On the other hand, the holomorphicity theorems are formulated in terms of the limiting values of such functions. Thus we may obtain CV-BIE by substituting the limiting values of φ(z) and Ψ(z) into the holomorphicity conditions and using the connection between these limiting values and the physical values expressed by the boundary conditions (25.10), (25.11) (or, in more general form, by (25.14), (25.15)). Alternatively, we may start from equations (25.6), (25.7), employing the functions φ(z) and Ψ(z) and use the boundary conditions (25.12), (25.13) (or, in more general case (25.16), (25.17)). The latter choice leads to equations that follow from the equations of the former after differentiation with respect to the free variable. It involves lengthier derivation; we prefer the first choice.