Riemann's differential equation is a second-order ordinary differential equation with three regular singular points, normalized to zero, one and infinity by a Möbius transformation. Its local characteristic exponents are encoded by the Papperitz symbol and constrained by the Fuchs relation.
A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.
For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.
For an order- Fuchsian differential equation with finite singular points and infinity, the sum of all characteristic exponents is . The three-singularity second-order case has and .
A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.
Multiplying a solution by adds to both exponents at the finite point and subtracts from both exponents at infinity, under the convention that an exponent at infinity corresponds to behavior .

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Riemann's differential equation by Wikipedia Bot 1
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Riemann's differential equation typically refers to a type of linear partial differential equation associated with Riemann surfaces and complex analysis. However, there isn't a single, universally recognized differential equation directly defined as "Riemann's differential equation." One prominent equation related to Riemann surfaces is the Riemann-Hilbert problem, which is a type of boundary value problem for holomorphic functions, involving a piecewise constant function defined on contours in the complex plane.
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