Abstract
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.
Similar content being viewed by others
References
Coulson C.A., O’Leary B., Mallion R.B. (1978). Hückel Theory for Organic Chemists. Academic Press, London
Yates K. (1978). Hückel Molecular Orbital Theory. Academic Press, New York
Graovac A., Gutman I., Trinajstić N. (1977). Topological Approach to the Chemistry of Conjugated Molecules. Springer-Verlag, Berlin
Dias J.R. (1993). Molecular Orbital Calculations Using Chemical Graph Theory. Springer-Verlag, Berlin
Coulson C.A. (1940). Proc. Camb. Phil. Soc. 36: 201
Coulson C.A., Longuet–Higgins H.C. (1947). Proc. Roy. Soc. Lond. A191: 39
Coulson C.A., Longuet–Higgins H.C. (1947). Proc. Roy. Soc. Lond. A192: 16
Coulson C.A., Longuet–Higgins H.C. (1948). Proc. Roy. Soc. Lond. A193: 447–456
Gutman I. (1977). J. Chem. Phys. 66: 1652
Gutman I. (1987). J. Math. Chem. 1: 123
Gutman I. (1992). Theory Chim. Acta 83: 313
Gutman I., Vidović D., Hosoya H. (2002). Bull. Chem. Soc. Jpn. 75: 1723
Gutman I., Milun M., Trinajstić N. (1977). J. Am. Chem. Soc. 99: 1692
Mizoguchi N. (1988). J. Phys. Chem. 92: 2754
Hosoya H., Hosoi K., Gutman I. (1975). Theory Chim. Acta 38: 37
Gutman I., Trinajstić N. (1976). J. Chem. Phys. 64: 4921
Bosanac S., Gutman I. (1977). Z. Naturforsch. 32a: 10
Gutman I., Polansky O.E. (1981). Theory Chim. Acta 60: 203
Mizoguchi N. (1989). Chem. Phys. Lett. 158: 383
Yvan P. (1952). J. Chim. Phys. 49: 457
Hosoya H., Hosoi K. (1976). J. Chem. Phys. 64: 1065
Gutman I., Yamaguchi T., Hosoya H. (1976). Bull. Chem. Soc. Jpn. 49: 1811
Hosoya H., Iwata S. (1999). Theory Chem. Acc. 102: 293
Mizoguchi N. (1984). Chem. Phys. Lett. 106: 451
Mizoguchi N. (1987). Bull. Chem. Soc. Jpn. 60: 2005
Gutman I. (1977). Theor. Chim. Acta 45: 79
Zhang J., Zhou B. (2005). J. Math. Chem. 37: 423
Gutman I., Polansky O.E. (1986). Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin
Hosoya H. (1999). J. Mol. Struct. (Theochem) 461/462: 473
Hosoya H. (2003). Bull. Chem. Soc. Jpn. 76: 2233
Gutman I. (2005). Monatsh. Chem. 136: 1055
O’Leary B., Mallion R.B. (1989). J. Math. Chem. 3: 323
Conway J. (1978). Functions of One Complex Variable. Springer-Verlag, Berlin
Churchill R., Brown J. (1984). Complex Variables and Applications. McGraw-Hill, New York
Barenstein A., Gay R. (1991). Complex Variables. Springer-Verlag, New York
Gutman I. (1979). Theory Chim. Acta 50: 287
Ilić P., Jurić A., Trinajstić N. (1980). Croat. Chem. Acta 53: 587
Author information
Authors and Affiliations
Corresponding author
Additional information
Ams Classification (2000): primary: 92E10, secondary: 32A27, 81Q99
Rights and permissions
About this article
Cite this article
Gutman, I., Mateljević, M. Note on the Coulson integral formula. J Math Chem 39, 259–266 (2006). https://doi.org/10.1007/s10910-005-9032-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-005-9032-2