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Linear Functional Equations with Several Independent Variables
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3. Linear Functional Equations with Several Independent Variables

f(x + y)=f(x) + f(y).
Cauchy's equation.

f(xy) = f(x) + f(y).
Cauchy's logarithmic equation.

2f(x + y) = f(2x) + f(2y).
Jensen's equation.

f(x + y) + f(x − y) = 2f(x) cosh y.

f(x + y) + f(x − y) = 2f(x) cos y.

f((x^{2} + y^{2})^{1/2}) = f(x)f(y).
Gauss equation.

f((x^{n} + y^{n})^{1/n}) = f(x) + f(y).

f(x) + g(y) = h(x + y).
Pexider's equation.

f(x) + (1 − x)f(y/(1 − x)) = f(y) + (1 − y)f(x/(1 − y)).
Equation of information theory.

f(1 − x) + (1 − x)^{α}f(y/(1 − x)) = f(y) + (1 − y)^{α}f(x/(1 − y)).

f(ax, ay) = f(x, y).

f(ax, ay) = a^{β}f(x, y).
Equation of homogeneous functions.

f(ax, a^{β}y) = f(x, y).

f(ax, a^{β}y) = a^{σ}f(x, y).
Equation of selfsimilar solutions.

f(x, y) + f(y, z) = f(x, z).
Cantor's first equation.
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