I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = \frac{(\theta_{11}(z;\tau))^2}{(\eta(\tau))^3} (\mu(z;t) - \mu(w;\tau))$$ where $\theta_{11}$ is one of the Jacobi theta functions, $$\theta_{11}(z;\tau) = i q^{1/8} y^{1/2} \prod\limits_{n=1}^{\infty} (1-q^n) (1-yq^n) (1-y^{-1}q^{n-1})$$ $\eta(\tau)$ is the Dedekind eta function $$\eta(\tau) = q^{1/24}\prod\limits_{n=1}^{\infty} (1-q^n)$$ and $\mu (z;\tau)$ is the following sum. $$\mu(z;\tau) = \frac{iy^{1/2}}{\theta_{11}(z;\tau)}\sum\limits_{n=1}^{\infty} (-1)^n \frac{q^{\frac{n(n+1)}{2}}y^n}{1-y q^n}$$ where $q = e^{2\pi i \tau}$ and $y = e^{2\pi i z}$ throughout, with $z \in \mathbb{C}$ and $\tau \in \mathbb{H}$.
It is argued below equation (3.22) that since the function $J(z;w;\tau)$ is holomorphic in all its variables, then fixing the value of $w$ to $\frac{1}{2}$, $\frac{1+\tau}{2}$ and $\frac{\tau}{2}$ gives us following identifications of J-function with ratios of theta functions based on the transformation properties of these functions:
$$J(z;\frac{1}{2};\tau) = \left(\frac{\theta_{10}(z;\tau)}{\theta_{10}(0;\tau)}\right)^2$$ $$J(z;\frac{1+\tau}{2};\tau) = \left(\frac{\theta_{00}(z;\tau)}{\theta_{00}(0;\tau)}\right)^2$$ $$J(z;\frac{\tau}{2};\tau) = \left(\frac{\theta_{01}(z;\tau)}{\theta_{01}(0;\tau)}\right)^2$$ My question is: Why is it so that holomorphicity and transformation properties of J-function at particular values of $w$ fixes the function completely?
In other words, how can the authors be sure that, say $J(z;\frac{1+\tau}{2};\tau)$ is exactly equal to $\left(\frac{\theta_{00}(z;\tau)}{\theta_{00}(0;\tau)}\right)^2$ based on the knowledge of the two functions' transformation properties? Can there not be two different holomorphic functions that transform in the same way?
For completeness, I present here the transformation properties of one of the examples above: $J(z;\frac{1+\tau}{2};\tau)$ and $\left(\frac{\theta_{00}(z;\tau)}{\theta_{00}(0;\tau)}\right)^2$.
$$J(z+1;\frac{1+\tau}{2};\tau) = J(z;\frac{1+\tau}{2};\tau+1) = J(z;\frac{1+\tau}{2};\tau)$$ $$J(z+\tau;\frac{1+\tau}{2};\tau) = q^{-1} y^{-2} J(z;\frac{1+\tau}{2};\tau)$$ $$J(\frac{z}{\tau};\frac{1+\tau}{2};\frac{-1}{\tau}) = e^{-2\pi i z^2 /\tau} J(z;\frac{1+\tau}{2};\tau)$$
and $\left(\frac{\theta_{00}(z;\tau)}{\theta_{00}(0;\tau)}\right)^2$ also has the same transformation properties.