Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals

Summary.

Starting with some results of Lyness concerning the Euler-Maclaurin expansion of Cauchy principal value integrals over $\left( 0,1\right)$ it is shown how, by the use of sigmoidal transformations, good approximations can be found for the Hadamard finite-part integral ${\int\!\!\!\! \scriptstyle{=}\;} _0^1\left (\phi \left( x\right) /\left( x-c\right) ^2\right) dx,$ where $0 < c < 1.$ The analysis is illustrated by some numerical examples.