Closed form solution for a hypersingular integral equation of order N+1

We consider the solution in closed form for a hypersingular integral equation of order n + 1 of the type phi (x) + (x-a)(1/2)(x-b)(1/2)/pii x integral (b)(a) phi (t)/(t-x)(n+1) dt = g(x), x is an element of (a,b) n = 1,2 ... . By using a complex function-theoretic method in a straightforward manner, after introducing a new unknown function, this equation is transformed into a differential Riemann boundary value problem on the cut (a, b), which is amenable to its closed form solution.