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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Riemann's differential equation, named after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
, is a generalization of the
hypergeometric differential equation In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
, allowing the regular singular points to occur anywhere on the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
, rather than merely at 0, 1, and \infty. The equation is also known as the Papperitz equation. The
hypergeometric differential equation In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
is a second-order linear differential equation which has three regular singular points, 0, 1 and \infty. That equation admits two linearly independent solutions; near a singularity z_s, the solutions take the form x^s f(x), where x = z-z_s is a local variable, and f is locally holomorphic with f(0)\neq0. The real number s is called the exponent of the solution at z_s. Let ''α'', ''β'' and ''γ'' be the exponents of one solution at 0, 1 and \infty respectively; and let ', ' and ' be those of the other. Then :\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1. By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
s will adjust the positions of the regular singular points, while other transformations (see below) can change the exponents at the regular singular points, subject to the exponents adding up to 1.


Definition

The differential equation is given by :\frac + \left \frac + \frac + \frac \right\frac ::+\left \frac +\frac +\frac \right\frac=0. The regular singular points are , , and . The exponents of the solutions at these regular singular points are, respectively, , , and . As before, the exponents are subject to the condition :\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.


Solutions and relationship with the hypergeometric function

The solutions are denoted by the ''Riemann P-symbol'' (also known as the ''Papperitz symbol'') :w(z)=P \left\ The standard
hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
may be expressed as :\;_2F_1(a,b;c;z) = P \left\ The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is :P \left\ = \left(\frac\right)^\alpha \left(\frac\right)^\gamma P \left\ In other words, one may write the solutions in terms of the hypergeometric function as :w(z)= \left(\frac\right)^\alpha \left(\frac\right)^\gamma \;_2F_1 \left( \alpha+\beta +\gamma, \alpha+\beta'+\gamma; 1+\alpha-\alpha'; \frac \right) The full complement of
Kummer Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chris ...
's 24 solutions may be obtained in this way; see the article
hypergeometric differential equation In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
for a treatment of Kummer's solutions.


Fractional linear transformations

The P-function possesses a simple symmetry under the action of
fractional linear transformation In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional t ...
s known as
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
s (that are the conformal remappings of the Riemann sphere), or equivalently, under the action of the group . Given arbitrary
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s , , , such that , define the quantities :u=\frac \quad \text \quad \eta=\frac and :\zeta=\frac \quad \text \quad \theta=\frac then one has the simple relation :P \left\ =P \left\ expressing the symmetry.


Exponents

If the Moebius transformation above moves the singular points but does not change the exponents, the following transformation does not move the singular points but changes the exponents: :\left(\frac\right)^k\left(\frac\right)^l P \left\ =P \left\


See also

* Method of Frobenius *
Monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...


Notes


References

* Milton Abramowitz and Irene A. Stegun, eds., '' Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'' (Dover: New York, 1972) *
Chapter 15
Hypergeometric Functions *

Riemann's Differential Equation {{Bernhard Riemann Hypergeometric functions Ordinary differential equations Bernhard Riemann