On Seiberg-Witten curves In page 44 of Gaiotto's article "Families of $\mathcal{N}=2$ Field Theories" on Teschner's review the author writes down the pure Seiberg-Witten curve as
$$
x^2 = z^3 + 2uz + \Lambda^4z
$$
with the SW differential being 
$$
\lambda = x \frac{dz}{z^2}
$$
On the other hand we have Tachikawa's review, in page 40, where he writes the Seiberg-Witten curve for the pure theory as
$$
\Lambda^2 z + \frac{\Lambda^2}{z} = x^2 - u
$$
with the SW differential being 
$$
\lambda = x \frac{dz}{z}
$$
To me the two SW curves look quite different so my question is what is map from one to another (if there exists one) and how one explaines the 2nd order pole in Gaiotto's curve? Even in the SW original paper the SW differential had a simple pole. 
Finally how does one go from either of the above to the original SW curve which is written as
$$
x^2 =  4z(z^2 - uz + \frac{1}{4}\Lambda^4) 
$$
 A: This doesn't require any understanding of SW theory. The different descriptions of the SW curve are related to each other by relabeling the variables. To make this clear, I'll rewrite the different descriptions using different subscripts. Gaiotto's version is
$$
 x_G^2 = z_G^3 + 2u_G z_G + \Lambda^4 z_G
\hskip1cm
 \lambda = x_G\frac{dz_G}{z_G^2}.
$$
Tachikawa's version is 
$$
 \Lambda^2 z_T + \frac{\Lambda^2}{z_T} = x_T^2-u_T
\hskip2cm
 \lambda = x_T\frac{dz_T}{z_T}.
$$
The original version is
$$
 x_{SW}^2 = 4z_{SW}
 \left(z_{SW}^2-u_{SW}z_{SW}+\frac{1}{4}\Lambda_{SW}^4\right).
$$
Substitute the relationships
$$
 x_T = \frac{x_G}{z_G}
\hskip1cm
 z_T = \frac{z_G}{\Lambda^2}
\hskip1cm
 u_T = 2u_G
$$
into Tachikawa's version and then multiply the whole equation by $z_G^2$ to get Gaiotto's version. Substitute the relationships
\begin{gather}
 x_{SW} = 2x_G
\hskip1cm
 z_{SW} = z_G
\\
 u_{SW} = -u_G
\hskip1cm
 \Lambda_{SW}^4 = 4\Lambda^4
\end{gather}
into the SW version to get Gaiotto's version.
