Abstract
A comparative study of the functional equationsf(x+y)f(x−y)=f 2(x)−f 2(y),f(y){f(x+y)+f(x−y)}=f(x)f(2y) andf(x+y)+f(x−y)=2f(x){1−2f 2(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x−y)=2f(x){1−2f 2(y/2)} with D'Alembert's equation,f(x+y)+f(x−y)=2f(x)f(y) and the sine-cosine equationg(x−y)=g(x)g(y) +f(x)f(y) has also been investigated.
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References
Aczél, J.,Lectures on functional equations and their applications. Academic Press, New York-London, 1966.
Cauchy, A. L.,Cours d'Analyse. Oeuvres, Ser 2.3 (1987), 106–113.
Flett, T.M.,Continuous solutions of the functional equation f(x + y) + f(x − y) = 2f(x)f(y). Amer. Math. Monthly70 (1963), 392–397.
Kannappan, Pl., The functional equation f(xy) + f(xy−1) = 2f(x)f(y) for groups. Proc. Amer. Math. Soc.29 (1968), 69–74.
Kurepa, S., On the functional equation f(x + y)f(x − y) = f2(x) − f2(y). Ann. Polon. Math.10 (1961), 1–5.
Kurepa, S.,On some functional equations in Banach spaces. Stud. Math. Appl.19 (1960), 149–158.
Rosenbaum, R. A. andSegal, S. L.,A functional equation characterising the sine. Math. Gaz.44 (1960), 97–105.
Segal, S. L.,On a sine functional equation. Amer. Math. Monthly70 (1963), 306–308.
Vaughan, H. E.,Characteristics of the sine and cosine. Amer. Math. Monthly62 (1955), 707–713.
Wilson, W. H.,Certain related functional equations. Bull. Amer. Math. Soc.26 (1919), 300–312.
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Parnami, J.C., Vasudeva, H.L. A comparative study of functional equations characterising sine and cosine. Aeq. Math. 31, 26–33 (1986). https://doi.org/10.1007/BF02188169
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DOI: https://doi.org/10.1007/BF02188169