The evaluation of Cauchy principal value integrals involving unknown poles (Q1822890)

If f(t) is analytic in [0,p] apart from a simple pole at an unknown point \(\lambda\) in (0,p) with an unknown residue \(\rho\), the authors consider, as the first author [Numer. Math. 21, 185-192 (1973; Zbl 0253.65011)], the analytic functions \(g_ n(t)=f(t)-\rho \phi_ n(t)\) where each \(\phi_ n(t)\) is a convenient periodic function. If: \(I(f)=\int^{P}_{0}f(t) dt,\) \(T_ n=\int^{P}_{0}g_ n(t) dt,\) the basic formula to evaluate I(f): \(I(f)=T_{2n}-(1/2)(T_{2n}-T_ n)^ 2(T_{2n}-T_{4n})^{-1}\) is obtained by elimination of \(\lambda\) and \(\rho\).