Reading Landau and Lifshitz "Quantum Mechanics. Non-relativistic theory", I've come across an identity, which after being a bit simplified, reads
and apparently holds for pure imaginary $a$ and $b$. The book says the left hand side can be calculated to equal right hand side using the reflection formula:
But I fail to see how I could actually do this calculation. Plotting the difference I see that this actually is false for real $a$ and $b$, so the reflection formula, which AFAIK works for all $x\in\mathbb C$ seems to be not enough to get this result.
So, how do I prove $(1)$, or even better, derive RHS from LHS?