Functional Equations and Applications-3 :
functional equations are given by the name of the mathematician who discovered it or sometimes functional equations are given names based on the property involved in the given functional equation.
Here is a list of well-known functional equations:
(i) Cauchy Functional Equation
f(x + y) = f(x) + f(y).
(ii) Jensen Functional Equation
F(( x + y)/2) = 1/2[f(x) + f(y)].
(iii) D'Alembert Functional Equation
f(x + y) + f(x − y) = 2f(x)f(y).
(iv) Wilson Functional Equation
g(x + y) + g(x − y) = 2g(x)f(y).
(v) Euler–Lagrange–Rassias Quadratic Functional Equation
f(ax + by) + f(bx − ay) = (a2 + b2)[f(x) + f(y)].
(vi) Abel Functional Equation
f(g(x)) = f(x) + α.
(vii) Pythagorean Functional Equation
|f(x + iy)|2 = |f(x)|2 + |f(iy)|2.
(viii) Davison Functional Equation
f(xy) + f(x + y) = f(xy + x) + f(y).
(ix) Sincov Functional Equation
ϕ (x, y) + ϕ (y, z) = ϕ (x, z).
(x) Riemann Functional Equation
ζ (1 - s) = Γ (s) / (2π) ^ s. 2 cos (πs /2)ζ (s).
(xi) Gauss Functional Equation
F(x2 + y2) = f(x)f(y).
(xii) Lobachevsky Functional Equation
f(x + y)f(x − y) = f2(x).
(xiii) Pompeiu Functional Equation
f(x + y + xy) = f(x) + f(y) + f(x)f(y).
(xiv) Drygas Functional Equation
f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y).
(xv) Swiatak Functional Equation
f(x + y) + f(x − y) = 2f(x) + 2f(y) + g(x)g(y).
(xvi) Hosszu Functional Equation
f(x + y − αxy) + g(xy) = h(x) + k(y).
(xvii) Baxter Functional Equation
f(f(x)y + f(y)x − xy) = f(x)f(y).
(xviii) Homogeneous Functional Equation
f(ax, ay) = aβf(x, y).
(xix) Associative Functional Equation
f(f(x, y), z) = f(x, f(y, z)).
(xx) Transitivity Functional Equation
f(x, y) = f(f(x, z), f(y, z)).
(xxi) Bisymmetry Functional Equation
f(f(x, y), f(u, z)) = f(f(x, u), f(y, z)).
(xxii) Cosine Functional Equations
(1) f(x + y) = f(x)f(y) − g(x)g(y),
(2) f(x − y) = f(x)f(y) + g(x)g(y).
(xxiii) (xxiii) Sinusoidal functional equations
(1) f(x + y) = f(x)g(y) + f(y)g(x),
(2) f(x − y) = f(x)g(y) − f(x)g(x).