Rimann differentsial tenglamasi

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

Ta'rif

Rieman differensial tenglamasi quyidagicha aniqlanadi

${\frac {d^{2}w}{dz^{2}}}+\left[{\frac {1-\alpha -\alpha '}{z-a}}+{\frac {1-\beta -\beta '}{z-b}}+{\frac {1-\gamma -\gamma '}{z-c}}\right]{\frac {dw}{dz}}$

$+\left[{\frac {\alpha \alpha '(a-b)(a-c)}{z-a}}+{\frac {\beta \beta '(b-c)(b-a)}{z-b}}+{\frac {\gamma \gamma '(c-a)(c-b)}{z-c}}\right]{\frac {w}{(z-a)(z-b)(z-c)}}=0.$

Uning muntazam yagona nuqtalari a, b va c bo'ladi. Ularning darajalari $\alpha$ va $\alpha '$, $\beta$ va $\beta '$, $\gamma$ va $\gamma '$ mos ravishda. Ular shartlarni qanoatlantiradi.

$\alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1.$

Tenglama yechimlari

Riman tenglamasining yechimlari Riman doim P belgisi bilan yoziladi

$w=P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}$

Odatiy gipergeometrik funktsiyani quyidagicha yozish mumkin bo'ladi

$\;_{2}F_{1}(a,b;c;z)=P\left\{{\begin{matrix}0&\infty &1&\;\\0&a&0&z\\1-c&b&c-a-b&\;\end{matrix}}\right\}$

P-funksiyalar bir qancha o'ziga xosliklarga bo'ysunadi, ulardan biri ularni gipergeometrik funksiyalar nuqtai nazaridan umumlashtirish imkonini beradi. Ya'ni,u ifoda quyidagicha

$P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}=\left({\frac {z-a}{z-b}}\right)^{\alpha }\left({\frac {z-c}{z-b}}\right)^{\gamma }P\left\{{\begin{matrix}0&\infty &1&\;\\0&\alpha +\beta +\gamma &0&\;{\frac {(z-a)(c-b)}{(z-b)(c-a)}}\\\alpha '-\alpha &\alpha +\beta '+\gamma &\gamma '-\gamma &\;\end{matrix}}\right\}$

shaklda tenglamaning yechimini yozish imkonini beradi

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

Mebius transformatsiyasi

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 $AD-BC\neq 0$, nisbatlarini aniqlang.

$u={\frac {Az+B}{Cz+D}}\quad {\text{ and }}\quad \eta ={\frac {Aa+B}{Ca+D}}$ va

$\zeta ={\frac {Ab+B}{Cb+D}}\quad {\text{ and }}\quad \theta ={\frac {Ac+B}{Cc+D}}.$

Keyingi tenglik

$P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}=P\left\{{\begin{matrix}\eta &\zeta &\theta &\;\\\alpha &\beta &\gamma &u\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}$

Adabiyotlar

• 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