New Approaches for the Real and Complex Integral Formulas of the Energy of a Polynomial

The energy of a graph was first defined in 1977. In 2010 (but also earlier) this concept was generalized to the energy of any complex polynomial. In this paper, we adopt new approaches to prove both the complex form and real form of the Coulson integral formulas for the energy of a complex polynomial. For the complex form, we use an approach which does not use the contour integration and the Cauchy residue theorem. For the real form, we use an approach which can completely avoid using the logarithm of a complex function. We also obtain the following new formula for the energy of an arbitrary monic complex polynomial φ(z): +∞ & −∞