I try to understand the answer to the following question as I want to deepen my knowledge about conformal mapping: Conformal mapping from triangle to upper half plane. I do not have enough knowledge and experience in this field yet to fully follow the answer but I have observed something puzzling: If I plot the proposed functions using Mathematica, neither of the two terms give the correct map.
Here is what the plots look like. (Of course, I can share the code if someone is interested.)
The function $f(z) = -\frac{1}{4e_1} \frac{(\wp(z)-e_1)^2}{\wp(z)}$ results in
while $f(z) = \frac{e_2-e_3}{(e_2-e_1)^2} \frac{(\wp(z)-e_1)^2}{\wp(z)-e_3}$ results in
.
That these two functions differ in the first place is not surprising as $e_3 = \wp(1+i) = \wp(1+i, 2, 2i) \approx -0.59i \neq 0$, according to Mathematica (in contrast to the claim in the answer).
I would be interested in how to fix the problem and to obtain the correct conformal mapping from the (Euclidean) triangle with vertices $0,i,1$ to the upper half-plane. I have looked into both the answer to the original question and further sources about the Weierstrass p-function but was not able to come up with a suggestion for the correct map. In particular, I have no intuition about $\wp$ and do not know if and how the mentioned derivative $\wp^\prime$ comes into play. Also, I would be grateful if someone could explain how to come up with the lattice generated by $2$ and $2i$.
Any explanations about the theoretical background as well as proposals for the correct mapping are highly appreciated! Thank you in advance.
EDIT: In the link cited in the other discussion, an explicit conformal map from an equilateral triangle to the upper half plane is given by $$\frac{1}{2} + \frac{27}{2 \beta(1/3,1/3)^3} \wp^\prime\left(z-e^{\pi i /3}, 0, \frac{- \beta(1/3,1/3)^6}{729}\right). $$ It would be great if anyone could share some insights where the $\wp^\prime$ and $\beta$ functions come from.
EDIT: The above plots are incorrect. Because I have no experience in this field, I mixed up the parameters $g_2,g_3$ with $\omega_1, \omega_2$. However, the Wikipedia article on the Weierstrass $\wp$ where the connection of the parameters is described is not clear to me at all. It would be very nice if someone could elaborate or refer me to a source which is readable for beginners.
I have read through different StackExchange and MathOverflow sites as well as the Mathematica documentation (where one can find WeierstrassInvariants and WeierstrassHalfPeriods which make $g$'s from the $\omega$'s and vice versa) where multiple definitions seem to exist. Could someone please explain what $g_1$ is what the exact relation between the $\omega$ is? (Are $\omega_1$ and $\omega_2$ a basis for the lattice or is it $\omega_1$ and $\omega_3$? What is the use of the additional $\omega_i$?)