We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If there exists a function $g(x)$ such that $f(x)g(x)$ is also integrable in $(a,b)$ and
$$ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx $$
then, $g(x)$ is the integral identity of $f(x)$ in the interval $(a,b)$.
Trivially the number 1 is an integral identity in all intervals. Given below is an example for a non trivial integral identity in the interval $(0,\infty)$. We have for $a > 0$ $$ \int_{0}^{\infty}\log\Big(1+\frac{a^2}{x^2}\Big)dx = \int_{0}^{\infty}\log\Big(1+\frac{a^2}{x^2}\Big)\log\Big(\frac{a}{x}\Big)dx = \pi a. $$
Questions:
- Is there any reference to such functions in literature?
- For what functions $f$ can we always find an integral identity in $(0,\infty)$ i.e how can we test if $f$ has an integral identity?
- For what functions $f$ can we always find an integral identity in all arbitrary intervals $(a,b)$?
- Can there be functions that have infinitely many integral identities in some intervals?