How can we obtain the expression of the function f : R --> R defined, for each (a, b) such that -oo < a <= b < +oo, by this equality?
Int(f(x), x=a..b) = Int(f(x), x=1/b..1/a)
Replacing x by 1/y in the rhs integral shows that f satisfies this functional equation
f(y) = f(1/y)/y^2
(note this impose y <> 0, see the PS below)
Functional equations is not y cup of tea and Maple (2015) doesn't seem to have any feature to solve them.
Could anyone explain me how f can be obtained?
I inadvertently obtained this equality between integrals as I was comparing the performances of different numerical integration methods.
One of the functions in my benchmarks suite had the expression f(x)=1/(1+x^2)... and I "discovered", by changing x to 1/y to ease the numerical integration, that f checked the inequality of the first code snippet.
Thus , for any constant C, C*f is a solution of the functional equation.
Note that f(x) being strictly positive, 1/y > 0 and the functional equation is always defined.