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.  

We have that
$$\tau'(-1/z)=-z^2\tau'(z)$$
The function $\sigma(z)=1/z$ has the following property:
$$\sigma(-1/z)=-z=-z^2\sigma(z)$$
So a solution could come from $\tau'(z)=1/z$. Then $\tau(z)=C+\log z$ where $C$ is a complex constant.
Then
$$C+\log(-1/z)=-C-\log(z)$$
$$C=-\frac{\log(z)+\log(-1/z)}2=-i\frac{\arg(z)+\pi-\arg(z)}2=-i\pi/2$$
Thus,
$$\tau(z)=-i\pi/2 + \log z=\log(-iz)$$
