Exact Solutions > Functional Equations > Linear Functional Equations with Several Independent Variables

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3. Linear Functional Equations with Several Independent Variables

1. f(x + y)=f(x) + f(y).    Cauchy's equation.
2. f(xy) = f(x) + f(y).    Cauchy's logarithmic equation.
3. 2f(x + y) = f(2x) + f(2y).    Jensen's equation.
4. f(x + y) + f(x − y) = 2f(x) cosh y.
5. f(x + y) + f(x − y) = 2f(x) cos y.   
6. f((x² + y²)^1/2) = f(x)f(y).    Gauss equation.
7. f((xⁿ + yⁿ)^1/n) = f(x) + f(y).
8. f(x) + g(y) = h(x + y).    Pexider's equation.
9. f(x) + (1 − x)f(y/(1 − x)) = f(y) + (1 − y)f(x/(1 − y)).    Equation of information theory.
10. f(1 − x) + (1 − x)^αf(y/(1 − x)) = f(y) + (1 − y)^αf(x/(1 − y)).
11. f(ax, ay) = f(x, y).
12. f(ax, ay) = a^βf(x, y).    Equation of homogeneous functions.
13. f(ax, a^βy) = f(x, y).
14. f(ax, a^βy) = a^σf(x, y).    Equation of self-similar solutions.
15. f(x, y) + f(y, z) = f(x, z).    Cantor's first equation.