**Riemann differensial tenglamasi** gipergeometrik tenglamani umumlashtirish bo'lib, u sizga muntazam yagona nuqtalar(ingl.)o'zb. olish imkonini beradi. ) Riman sferasining istalgan nuqtasida. Matematik Bernxard Rimann sharafiga nomlangan.

Ta'rif
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Rieman differensial tenglamasi quyidagicha aniqlanadi

{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left[{\frac {1-\alpha -\alpha '}{z-a}}+{\frac {1-\beta -\beta '}{z-b}}+{\frac {1-\gamma -\gamma '}{z-c}}\right]{\frac {dw}{dz}}}
{\displaystyle +\left[{\frac {\alpha \alpha '(a-b)(a-c)}{z-a}}+{\frac {\beta \beta '(b-c)(b-a)}{z-b}}+{\frac {\gamma \gamma '(c-a)(c-b)}{z-c}}\right]{\frac {w}{(z-a)(z-b)(z-c)}}=0.}
Uning muntazam yagona nuqtalari *a*, *b* va *c* bo'ladi. Ularning darajalari {\displaystyle \alpha } va {\displaystyle \alpha '}, {\displaystyle \beta } va {\displaystyle \beta '}, {\displaystyle \gamma } va {\displaystyle \gamma '} mos ravishda. Ular shartlarni qanoatlantiradi.

{\displaystyle \alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1.}

Tenglama yechimlari
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Riman tenglamasining yechimlari Riman doim *P belgisi* bilan yoziladi

{\displaystyle w=P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}}
Odatiy gipergeometrik funktsiyani quyidagicha yozish mumkin bo'ladi

{\displaystyle \;\_{2}F\_{1}(a,b;c;z)=P\left\{{\begin{matrix}0&\infty &1&\;\\0&a&0&z\\1-c&b&c-a-b&\;\end{matrix}}\right\}}
P-funksiyalar bir qancha o'ziga xosliklarga bo'ysunadi, ulardan biri ularni gipergeometrik funksiyalar nuqtai nazaridan umumlashtirish imkonini beradi. Ya'ni,u ifoda quyidagicha

{\displaystyle P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}=\left({\frac {z-a}{z-b}}\right)^{\alpha }\left({\frac {z-c}{z-b}}\right)^{\gamma }P\left\{{\begin{matrix}0&\infty &1&\;\\0&\alpha +\beta +\gamma &0&\;{\frac {(z-a)(c-b)}{(z-b)(c-a)}}\\\alpha '-\alpha &\alpha +\beta '+\gamma &\gamma '-\gamma &\;\end{matrix}}\right\}}
shaklda tenglamaning yechimini yozish imkonini beradi

{\displaystyle w=\left({\frac {z-a}{z-b}}\right)^{\alpha }\left({\frac {z-c}{z-b}}\right)^{\gamma }\;\_{2}F\_{1}\left(\alpha +\beta +\gamma ,\alpha +\beta '+\gamma ;1+\alpha -\alpha ';{\frac {(z-a)(c-b)}{(z-b)(c-a)}}\right)}

Mebius transformatsiyasi
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p-funksiya Mebius o'zgarishiga nisbatan oddiy simmetriyaga ega, ya'ni GL(2, **C** ) yoki ekvivalenti bilan Riman sferasining konformal xaritasi deyiladi. O'zboshimchalik bilan tanlangan to'rtta kompleks sonlar *A*, *B*, *C* va *D* shartni qondiradi {\displaystyle AD-BC\neq 0}, nisbatlarini aniqlang.

{\displaystyle u={\frac {Az+B}{Cz+D}}\quad {\text{ and }}\quad \eta ={\frac {Aa+B}{Ca+D}}} va
{\displaystyle \zeta ={\frac {Ab+B}{Cb+D}}\quad {\text{ and }}\quad \theta ={\frac {Ac+B}{Cc+D}}.}
Keyingi tenglik

{\displaystyle 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\}}

Adabiyotlar
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* Milton Abramowitz va Irene A. Stegun, muharrirlar *, Formulalar, grafiklar va matematik jadvallar bilan matematik funktsiyalar bo'yicha qo'llanma* (Dover: Nyu-York, 1972)
	+ 15-bob Gipergeometrik funksiyalar
		- 15.6-bo'lim Rimanning differentsial tenglamasi