1. ## f(x^2+y^2)=g(x)g(y) ...

Hello everybody!

A problem:

Find functions $\displaystyle \varphi , \psi$ which are fulfilling:

$\displaystyle \varphi(x^2+y^2)=\psi(x) \psi(y)$ for all $\displaystyle x,y$.

Prove that if $\displaystyle \varphi , \psi$ are fulfilling the above equation then $\displaystyle \psi$ determined by $\displaystyle \varphi$. How?

Thank you.

2. For example $\displaystyle \phi=\psi =0$. A more interesting example is given by $\displaystyle \phi :x\mapsto e^x$ and $\displaystyle \psi:x\mapsto e^{x^2}$.

3. Originally Posted by girdav
For example $\displaystyle \phi=\psi =0$. A more interesting example is given by $\displaystyle \phi :x\mapsto e^x$ and $\displaystyle \psi:x\mapsto e^{x^2}$.

What you wrote and the family of functions: a^x where a>1 ?

What about the second part of the question?

Thanks!

4. This is how I might attempt to do this problem. I will assume that both $\displaystyle \psi$ and $\displaystyle \phi$ are smooth. Taking the natural log of both sides gives

$\displaystyle \ln \phi(x^2+y^2) = \ln \psi(x) + \ln \psi(y)$.

Call the term on the RHS $\displaystyle F(x^2+y^2)$. Differentiating both side wrt $\displaystyle x$ and $\displaystyle y$ gives

$\displaystyle F''(x^2+y^2) = 0$.

Thus, $\displaystyle F(x^2+y^2) = a(x^2+y^2) + \ln(b)$ where $\displaystyle a$ and $\displaystyle b$ are constant.

So $\displaystyle \ln \phi(x^2+y^2) = a(x^2+y^2) + \ln(b)$

so $\displaystyle \phi(x^2+y^2) = b e^{a(x^2+y^2)} = \psi(x)\psi(y)$.

Now set $\displaystyle y = 0$ and this gets you the form for $\displaystyle \psi(x) = k e^{ax^2}.$ Then substitute into the original functional equation to determine $\displaystyle k$.

5. Originally Posted by Also sprach Zarathustra
Hello everybody!

A problem:

Find functions $\displaystyle \varphi , \psi$ which are fulfilling:

$\displaystyle \varphi(x^2+y^2)=\psi(x) \psi(y)$ for all $\displaystyle x,y$.

Prove that if $\displaystyle \varphi , \psi$ are fulfilling the above equation then $\displaystyle \psi$ determined by $\displaystyle \varphi$. How?

Thank you.
Maybe there is something missing from the problem statement. Otherwise
$\displaystyle \phi(x) = e^x$ and
$\displaystyle \psi(x) = - e^{x^2}$,
combined with girdav's solution, provides a counterexample to the second part (uniqueness).