Implement Mobius Transform and Its Inverse in Arb4J

[crowlogic]

Implement Möbius Transform and Its Inverse in Arb4J

Objective

Implement the Möbius transformation and its inverse in Arb4J, the arbitrary precision SWIG wrapper around the Arb library for Java.

Description

• Möbius Transformations are complex functions of the form $f(z) = \frac{az + b}{cz + d}$, where $a, b, c,$ and $d$ are complex numbers, and $z$ is a complex variable.
• These transformations can be represented using 2x2 matrices with complex entries.
• The vector $[z, 1]$ in these transformations can be interpreted as a first-degree polynomial $z + 1$.

Task Details

1. Polynomial Representation:

   • Understand the vector $[z, 1]$ as a first-degree polynomial, specifically $P(z) = z + 1$.
   • This interpretation is crucial for operations in projective spaces and complex analysis.

2. Matrix Representation:

   • Represent a Möbius transformation using a 2x2 matrix with entries $[a, b], [c, d]$.
   • For example, for $f(z) = \frac{2z + 3}{4z + 5}$, use the matrix

$$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$

3. Implementation:

   • Support complex arithmetic and polynomial representations in the implementation.
   • Implement functionality to apply the Möbius transformation to polynomials representing complex numbers, particularly $z + 1$.

4. Inverse Transformation:

   • Implement the inverse of the Möbius transformation.
   • The inverse is given by $f^{-1}(z) = \frac{dz - b}{-cz + a}$ for the matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$.

Testing

• Develop unit tests to verify both the Möbius transformation and its inverse.
• Include tests for polynomial representations, edge cases, and typical use cases.

Documentation

• Clearly document the usage of the implementation.
• Provide examples demonstrating the transformation and its inverse in the context of polynomial representations.

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This feature will significantly enhance the mathematical capabilities of Arb4J, making it a more versatile tool for complex computations in Java.

[crowlogic]

dont care enough righy now