One of the applications of the holomorphic functional calculus is with regard to idempotents. For instance, if an element $a$ in a unital Banach algebra $A$ has spectrum contained in two balls, each of radius $\frac{1}{4}$ and centered at 0 and at 1 respectively, then by taking a holomorphic function that takes value 0 on the ball around 0 and takes value 1 on the ball around 1, one can apply the holomorphic functional calculus to produce an idempotent in $A$.

My question is: What happens if the original element $a$ was already an idempotent? Do we get $a$ itself after going through the same process?

I've looked at a few books that mention this application of the functional calculus but none of them mention this special case and I haven't been able to prove it myself. I hope it doesn't turn out to be a silly question.