Difference and functional equations Definition

Cauchy's power equation

Definition

Cauchy's power equation is the functional equation $f(xy) = f(x)\, f(y) \quad \text{for all } x, y > 0,$ for an unknown function $f \colon (0, \infty) \to \R$ . A solution is a function $f$ that satisfies this identity for all positive $x$ and $y$ .

Remarks

Example: $f(x) = \sqrt{x}$ is a solution, since $\sqrt{xy} = \sqrt{x} \sqrt{y}$ . Non-example: $f(x) = 2x$ is not a solution, since $f(1 \cdot 1) = 2$ while $f(1) f(1) = 4$ . The substitution $x = e^s$ , $y = e^t$ turns the equation into Cauchy's exponential equation for $g(s) = f(e^s)$ . So the continuous solutions are $f = 0$ and $f(x) = x^c$ with $c \in \R$ .

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Tag	`STJAXZG1K1`
URI	`m2://STJAXZG1K1`
Kind	definition
Writing	W0 stub
MSC	39B22
Defines	Cauchy's power equation
Uses	Functional equation, Function

Cauchy's power equation

Definition

Remarks

Reading path

Further reading

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