The integral sine addition law.

  • D. Zeglami Moulay Ismail University.
  • M. Tial Ibn Tofail University.
  • S. Kabbaj Ibn Tofail University.
Keywords: Functional equation, Sine and cosine addition laws, Character, Borel measure


In the present paper we determine, in terms of characters and additive functions, the solutions of the integral functional equation for the sine addition law  ∫ G f(xyt)dµ(t) = f(x)g(y) + g(x)f(y), x, y ∈ G, where G is a locally compact Hausdorff group and µ is a regular, compactly supported, complex-valued Borel measure on G. Some consequences of this result and an example are presented.

Author Biographies

D. Zeglami, Moulay Ismail University.
Department of Mathematics, ENSAM.
M. Tial, Ibn Tofail University.
Department of Mathematics, Faculty of Sciences.
S. Kabbaj, Ibn Tofail University.
Department of Mathematics, Faculty of Sciences.


J. Aczél, Lectures on Functional Equations and Their Applications. Mathematics in Science and Engineering, vol. 19. Academic Press, New York, xx+510, (1966).

J. K. Chung, Pl. Kannappan, C. T. Ng, A generalization of the cosinesine functional equation on groups. Linear Algebra Appl. 66, pp. 259-277, (1985).

B. R. Ebanks, H. Stetkær, d’Alembert’s other functional equation on monoids with an involution. Aequationes Math. 89 (1), pp. 187-206, (2015).

B. Fadli, D. Zeglami, S. Kabbaj, The generalized Van Vleck’s equation on locally compact groups. Proyecciones J. of Math., 36 (4), pp. 545-566, (2017).

B. Fadli, D. Zeglami, S. Kabbaj, An integral functional equation on groups under two measures. Proyecciones J. of Math., 37 (3), pp. 565-581, (2018).

Pl. Kannappan, Functional Equations and Inequalities with Applications. Springer, New York, 39-02 (39Bxx), (2009).

Th. A. Poulsen, H. Stetkær, On the trigonometric subtraction and addition formulas. Aequationes Math. 59 (1), pp. 84-92 (2000).

H. Stetkær, ”Functional equations on groups”. World Scientific Publishing Company, Singapore, (2013).

H. Stetkær, Van Vleck’s functional equation for the sine. Aequationes Math. 90 (1), pp. 25—34, (2016).

H. Stetkær, The cosine addition law with an additional term, Aequationes Math. 90 (6), pp. 1147-1168, (2016).

L. Székelyhidi, Convolution Type Functional Equations on Topological Abelian Groups, World Scientific Publishing Company, Singapore-New Jersey-London-Hong Kong, (1991).

E. B. Van Vleck, A functional equation for the sine. Ann. of Math., Second Series, 11 (4), pp. 161-165, (1910).

D. Zeglami, B. Fadli, S. Kabbaj, Harmonic analysis and generalized functional equations for the cosine, Adv. Pure Appl. Math. 7 (1), pp. 41-49, (2016).

D. Zeglami, B. Fadli, Integral functional equations on locally compact groups with involution, Aequationes Math. 90 (5), pp. 967-982, (2016).

D. Zeglami, Some functional equations related to number theory, Acta Math. Hungar. 149, no. 2, pp. 490-508, (2016).

D. Zeglami, M. Tial, S. Kabbaj, The integral cosine addition and sine subtraction laws, Results Math. 73, no.3, 97, (2018).

How to Cite
D. Zeglami, M. Tial, and S. Kabbaj, “The integral sine addition law.”, Proyecciones (Antofagasta, On line), vol. 38, no. 2, pp. 203-219, May 2019.