Note on the Coulson integral formula

Let P be a polynomial of degree n, whose zeros λ₁, λ₂, ..., λ _n are real-valued. The Coulson integral formula (first reported in 1940) is an expression for the sum of the positive–valued zeros of P, in terms of P. We show that the Coulson formula holds if and only if the condition λ₁+λ₂+...+λ _n =0 is obeyed. We also show how the formula has to be modified, so that it be applicable in the case when λ₁+λ₂+...+λ _n ≠ 0. Ams Classification (2000): primary: 92E10, secondary: 32A27, 81Q99