How do i find a function such as #f(a)f(b)f(c) = f(sqrt(a^2+b^2+c^2))f^2(0)# ?
1 Answer
#f(x) = p k^(x^2)# for constants#p in RR# ,#k > 0#
Explanation:
I will assume that by
f(x) is an even function
First note that if
Otherwise, if we let
#f(a)f(0)f(0) = f(sqrt(a^2+0^2+0^2))f(0)f(0)#
and hence:
#f(a) = f(sqrt(a^2)) = f(abs(a))#
So we can deduce that
Any constant function is a solution
Suppose
Then:
#f(a)f(b)f(c) = k^3 = f(sqrt(a^2+b^2+c^2))f(0)f(0)#
Are there any non-constant solutions?
Suppose
Notice that:
#f(sqrt(2)) = f(sqrt(1^2+1^2+0^0))f(0)f(0) = f(1)f(1)f(0) = k^2#
#f(sqrt(3)) = f(sqrt(1^2+1^2+1^2))f(0)f(0) = f(1)f(1)f(1) = k^3#
Observing this pattern, we can define
#f(x) = k^(x^2)#
To find:
#f(a)f(b)f(c) = k^(a^2)k^(b^2)k^(c^2)#
#= k^(a^2+b^2+c^2)#
#= k^((sqrt(a^2+b^2+c^2))^2)#
#= f(sqrt(a^2+b^2+c^2))#
#= f(sqrt(a^2+b^2+c^2))f(0)f(0)#
Note that if
#f(x) = p k^(x^2)#
will also be a solution.
The case