I am confused about the behavior of the Stieltjes transform $s_\mu(x)$ for an even measure $d\mu(x)$. Let $d\mu(x)$ be positive and even on $[-1,1]$; the Stieltjes transform is the function $$ s_\mu(z) = \int_{-1}^1 \frac{d\mu(x)}{z-x} $$ Naively, if $d\mu(x) = d\mu(-x)$, I would make the substitution $s = -x$ to show $$ s_\mu(-z) = -\int_{-1}^1 \frac{d\mu(x)}{z+x} = -\int_{s = 1}^{s=-1} \frac{d\mu(-s)}{z-s} = \int_{-1}^1 \frac{d\mu(s)}{z-s} = s_\mu(z) $$ suggesting that for an even measure $d\mu(x)$ the Stieltjes transform is invariant under the involution $z \to -z$. However, looking at explicit examples this doesn't seem to be the case. For example, the semicircle measure $\sim \sqrt{1-x^2}$ gives $s(z) \sim z - \sqrt{z^2- 1}$.
I expect my error is simple but I'm not able to find it on my own. What, if anything, is the relation of $s(z),s(-z)$ for an even measure? What is wrong with the manipulations above?