Hello everybody!

A problem:

Find functions which are fulfilling:

for all .

Prove that if are fulfilling the above equation then determined by . How?

Thank you.

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- May 26th 2011, 04:18 AMAlso sprach Zarathustraf(x^2+y^2)=g(x)g(y) ...
Hello everybody!

A problem:

Find functions which are fulfilling:

for all .

Prove that if are fulfilling the above equation then determined by . How?

Thank you. - May 26th 2011, 04:26 AMgirdav
For example . A more interesting example is given by and .

- May 26th 2011, 04:59 AMAlso sprach Zarathustra
- May 27th 2011, 07:45 AMJester
This is how I might attempt to do this problem. I will assume that both and are smooth. Taking the natural log of both sides gives

.

Call the term on the RHS . Differentiating both side wrt and gives

.

Thus, where and are constant.

So

so .

Now set and this gets you the form for Then substitute into the original functional equation to determine . - May 29th 2011, 02:05 PMawkward