Functional equation

1. Dec 13, 2009

zenos

1. The problem statement, all variables and given/known data

Is the solution correct

2. Relevant equations

3. The attempt at a solution

all are in the file

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• file.pdf
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Last edited: Dec 13, 2009
2. Dec 13, 2009

snipez90

That does not look like a complete solution. For one, f(x) = 1 for all x and f identically equal to 0 are trivial solutions, and these can be cited by inspection. I'll keep trying things, but it would help if there were any additional assumptions on f, such as continuity perhaps?

3. Dec 13, 2009

snipez90

All right, here is a rough sketch for the case where f is continuous. As before, f identically equal to zero is a trivial solution. Now suppose there exists a real number c for which f(c) =/= 0. Then

$$f(x)f(c) = f(\sqrt{x^2 + c^2}) = f(-x)f(c).$$

This implies that f(x) = f(|x|) for all real x. Define $g(x) = f(\sqrt{x})$ for $x \geq 0$. Note that g satisfies Cauchy's exponential equation: g(x + y) = g(x)g(y) for $x,y \geq 0.$

Now see if you can complete the argument based off of the proof for Cauchy's exponential equation. For reference, attached is something I wrote awhile ago when I was still interested in functional equations.

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