Differential Equation
The Schwarzian derivative has a fundamental relation with a second-order linear ordinary differential equation in the complex plane. Let and be two linearly independent holomorphic solutions of
Then the ratio satisfies
over the domain on which and are defined, and The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions and can be found, and furthermore, these are unique up to a common scale factor.
When a linear second-order ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Q-value of the equation.
Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.
Read more about this topic: Schwarzian Derivative
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