Conformal Mapping between the Circle and the Square

The theory of Schwarz-Christoffel transformations in complex analysis provides a conformal way of mapping a circular disc to a square and vice versa. Here is a fundamental diagram for this mapping in the complex plane.

Conformal map from unit circular disc to square & vice versa

□→ ◯

To map from square to circular disc, one needs the complex-valued Jacobi elliptic function cn.
http://mathworld.wolfram.com/JacobiEllipticFunctions.html

◯ → □

To map from circular disc to square, one needs the complex-valued Legendre elliptic integral of the 1st kind F.

http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html

For a mathematical derivation of this diagram, see Section 8 of my paper in 

https://arxiv.org/ftp/arxiv/papers/1509/1509.06344.pdf

It is relatively easy to verify this diagram through Mathematica. One can also do it online through Wolfram Alpha. Mathematica has built-in functions JacobiCN and EllipticF that can take-in complex-valued inputs.

https://www.wolframalpha.com/input/?i=JacobiCN%5B-1.854%2B1.854*I%2C1%2F2%5D

Using the Jacobi cn function in Wolfram Alpha

Unfortunately, this fundamental diagram  has the square off-center and tilted by 45º . For many applications, it is preferable to have the square centered and oriented correctly. My previous blog entry has equations for this
https://squircular.blogspot.com/2015/09/schwarz-christoffel-mapping.html