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I need to solve the following functional equation: $xf\left(\frac{1}{x}\right)=f(x)$.

By observation, $f(x)=\sqrt{x}$ is a solution, but I don't know how to find a general solution, or show that this is the only solution. I've tried forming a differential equation, and just trying out different values for x, but so far have nothing.

Ollie
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1 Answers1

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Let $g(x)=f(x)/\sqrt{x}$ and we have $g(x)=g(1/x)$. We only need to consider $g$.

However, $g$ is free on $[1,+\infty)$, which means you can randomly set the value $g(x)$ there and just extend it onto $(0,1)$ according to the equation above.

rushusuixing
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    Another way to phrase this requirement is that $h(x) \equiv g(e^{x})$ must be an even function, or in other words $f$ is of the form $f(x) = \sqrt{x} h(\log x)$ for even function $h(x)$ – Ninad Munshi Dec 18 '23 at 16:34