Riemann's Differential Equation - Fractional Linear Transformations

Fractional Linear Transformations

The P-function possesses a simple symmetry under the action of fractional linear transformations, that is, under the action of the group GL(2, C), or equivalently, under the conformal remappings of the Riemann sphere. Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities

u=\frac{Az+B}{Cz+D} \quad \text{ and } \quad \eta=\frac{Aa+B}{Ca+D}

and

\zeta=\frac{Ab+B}{Cb+D} \quad \text{ and } \quad \theta=\frac{Ac+B}{Cc+D}

then one has the simple relation

P \left\{ \begin{matrix} a & b & c & \; \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' & \; \end{matrix} \right\} =P \left\{ \begin{matrix} \eta & \zeta & \theta & \; \\ \alpha & \beta & \gamma & u \\ \alpha' & \beta' & \gamma' & \; \end{matrix} \right\}

expressing the symmetry.

Read more about this topic:  Riemann's Differential Equation

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