Lerch's Theorem

If there are two functions $F_1(t)$ and $F_2(t)$ with the same integral transform

$${\mathcal T}[F_1(t)] = {\mathcal T}[F_2(t)] \equiv f(s),$$ (1)

then a Null Function can be defined by

$$\delta_0(t) \equiv F_1(t)-F_2(t)$$ (2)

so that the integral

$$\int^a_0 \delta_0(t)\,dt = 0$$ (3)

vanishes for all $a > 0$.

See also Null Function