Solving the functional equation $\tau \left(\frac{-1}{z}\right) = - \tau(z)$ 
Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. Find $\tau: \mathbb{H} \to \mathbb{C} $, holomorphic and non-constant, satisfying $\tau \left( \frac{-1}{z} \right) = - \tau(z)$. 

There is a very good answer already. But since the question was put on hold, I'll add some context. This question came up in the context of Klein's $j$ - invariant. If you change coordinates from $z \to w=z-i$ and expand in $w$, you'll see from symmetry arguments that the first two coefficients must be $0$. From this, I'm wondering if $j$ is locally the square of a coordinate around $i$. This makes sense if you look at the fundamental domain for $\mathbb{H}$ mod the $SL(2, \mathbb{Z})$ action, and note that $j$ defines a complex analytic structure on it.  

EqWorld is our right partner.
In fact this functional equation belongs to the form of http://eqworld.ipmnet.ru/en/solutions/fe/fe1121.pdf.
The general solution is $\tau(z)=C\left(z~,-\dfrac{1}{z}\right)$ , where $C(u,v)$ is any antisymmetric function.
