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Rimann differentsial tenglamasi

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

Ta'rif

Rieman differensial tenglamasi quyidagicha aniqlanadi

d2wdz2+[1−α−α′z−a+1−β−β′z−b+1−γ−γ′z−c]dwdz{\displaystyle \frac{d^2w}{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}}

+[αα′(a−b)(a−c)z−a+ββ′(b−c)(b−a)z−b+γγ′(c−a)(c−b)z−c]w(z−a)(z−b)(z−c)=0.{\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 α{\displaystyle \alpha} va α′{\displaystyle \alpha'}, β{\displaystyle \beta} va β′{\displaystyle \beta'}, γ{\displaystyle \gamma} va γ′{\displaystyle \gamma'} mos ravishda. Ular shartlarni qanoatlantiradi.

α+α′+β+β′+γ+γ′=1.{\displaystyle \alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.}

Tenglama yechimlari

Riman tenglamasining yechimlari Riman doim P belgisi bilan yoziladi

w=P{abcαβγzα′β′γ′}{\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

2F1(a,b;c;z)=P{0∞10a0z1−cbc−a−b}{\displaystyle \;_2F_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{abcαβγzα′β′γ′}=(z−az−b)α(z−cz−b)γP{0∞10α+β+γ0(z−a)(c−b)(z−b)(c−a)α′−αα+β′+γγ′−γ}{\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

w=(z−az−b)α(z−cz−b)γ2F1(α+β+γ,α+β′+γ;1+α−α′;(z−a)(c−b)(z−b)(c−a)){\displaystyle w= \left(\frac{z-a}{z-b}\right)^\alpha \left(\frac{z-c}{z-b}\right)^\gamma \;_2F_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≠0{\displaystyle AD-BC \neq 0}, nisbatlarini aniqlang.

u=Az+BCz+Dandη=Aa+BCa+D{\displaystyle u=\frac{Az+B}{Cz+D} \quad \text{ and } \quad \eta=\frac{Aa+B}{Ca+D}} va

ζ=Ab+BCb+Dandθ=Ac+BCc+D.{\displaystyle \zeta=\frac{Ab+B}{Cb+D} \quad \text{ and } \quad \theta=\frac{Ac+B}{Cc+D}.}

Keyingi tenglik

P{abcαβγzα′β′γ′}=P{ηζθαβγuα′β′γ′}{\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

  • Milton Abramowitz va Irene A. Stegun, muharrirlar , Formulalar, grafiklar va matematik jadvallar bilan matematik funktsiyalar bo'yicha qo'llanma (Dover: Nyu-York, 1972) 
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