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| In [[mathematics]], '''Riemann's differential equation''', named after [[Bernhard Riemann]], is a generalization of the [[hypergeometric differential equation]], allowing the [[regular singular points]] to occur anywhere on the [[Riemann sphere]], rather than merely at 0, 1, and ∞.
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| ==Definition==
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| The differential equation is given by
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| :<math>\frac{d^2w}{dz^2} + \left[
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| \frac{1-\alpha-\alpha'}{z-a} +
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| \frac{1-\beta-\beta'}{z-b} +
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| \frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz} </math>
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| ::<math>+\left[
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| \frac{\alpha\alpha' (a-b)(a-c)} {z-a}
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| +\frac{\beta\beta' (b-c)(b-a)} {z-b}
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| +\frac{\gamma\gamma' (c-a)(c-b)} {z-c}
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| \right]
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| \frac{w}{(z-a)(z-b)(z-c)}=0.</math>
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| The regular singular points are {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. The pairs of exponents{{clarification needed|date=November 2013|reason=There is no exponentiation yet, and you may not call it “exponents”}} for each are respectively {{math|''α''; ''α′''}}, {{math|''β''; ''β′''}}, and {{math|''γ''; ''γ′''}}. The exponents are subject to the condition
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| :<math>\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.</math>
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| ==Solutions==
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| The solutions are denoted by the ''Riemann P-symbol''
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| :<math>w(z)=P \left\{ \begin{matrix} a & b & c & \; \\
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| \alpha & \beta & \gamma & z \\
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| \alpha' & \beta' & \gamma' & \;
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| \end{matrix} \right\}</math>
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| The standard [[hypergeometric function]] may be expressed as
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| :<math>\;_2F_1(a,b;c;z) =
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| P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\
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| 0 & a & 0 & z \\
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| 1-c & b & c-a-b & \;
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| \end{matrix} \right\}</math>
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| 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
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| :<math>P \left\{ \begin{matrix} a & b & c & \; \\
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| \alpha & \beta & \gamma & z \\
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| \alpha' & \beta' & \gamma' & \;
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| \end{matrix} \right\} =
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| \left(\frac{z-a}{z-b}\right)^\alpha
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| \left(\frac{z-c}{z-b}\right)^\gamma
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| P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\
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| 0 & \alpha+\beta+\gamma & 0 & \;\frac{(z-a)(c-b)}{(z-b)(c-a)} \\
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| \alpha'-\alpha & \alpha+\beta'+\gamma & \gamma'-\gamma & \;
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| \end{matrix} \right\}
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| </math>
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| In other words, one may write the solutions in terms of the hypergeometric function as
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| :<math>w(z)=
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| \left(\frac{z-a}{z-b}\right)^\alpha
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| \left(\frac{z-c}{z-b}\right)^\gamma
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| \;_2F_1 \left(
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| \alpha+\beta +\gamma,
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| \alpha+\beta'+\gamma;
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| 1+\alpha-\alpha';
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| \frac{(z-a)(c-b)}{(z-b)(c-a)} \right)
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| </math>
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| The full complement of [[Ernst Kummer|Kummer]]'s 24 solutions may be obtained in this way; see the article [[hypergeometric differential equation]] for a treatment of Kummer's solutions.
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| ==Fractional linear transformations==
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| The P-function possesses a simple symmetry under the action of [[fractional linear transformation]]s known as [[Möbius transformation]]s (that are the [[conformal map|conformal remappings]] of the Riemann sphere), or equivalently, under the action of the group {{math|[[general linear group|GL]](2, '''C''')}}. Given arbitrary [[complex number]]s {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, {{mvar|D}} such that {{math|''AD'' − ''BC'' ≠ 0}}, define the quantities
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| :<math>u=\frac{Az+B}{Cz+D}
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| \quad \text{ and } \quad
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| \eta=\frac{Aa+B}{Ca+D}</math>
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| and
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| :<math>\zeta=\frac{Ab+B}{Cb+D} | |
| \quad \text{ and } \quad
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| \theta=\frac{Ac+B}{Cc+D}</math>
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| then one has the simple relation
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| :<math>P \left\{ \begin{matrix} a & b & c & \; \\
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| \alpha & \beta & \gamma & z \\
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| \alpha' & \beta' & \gamma' & \;
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| \end{matrix} \right\}
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| =P \left\{ \begin{matrix}
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| \eta & \zeta & \theta & \; \\
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| \alpha & \beta & \gamma & u \\
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| \alpha' & \beta' & \gamma' & \;
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| \end{matrix} \right\}</math>
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| expressing the symmetry.
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| ==See also==
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| *[[Complex differential equation]]
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| ==References==
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| * Milton Abramowitz and Irene A. Stegun, eds., ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'' (Dover: New York, 1972)
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| ** [http://www.math.sfu.ca/~cbm/aands/page_556.htm Chapter 15] Hypergeometric Functions
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| ***[http://www.math.sfu.ca/~cbm/aands/page_564.htm Section 15.6] Riemann's Differential Equation
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| [[Category:Hypergeometric functions]]
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| [[Category:Ordinary differential equations]]
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