Rimann differentsial tenglamasi

Riemann differensial tenglamasi gipergeometrik tenglamani umumlashtirish bo'lib, u sizga Andoza:Tarjima qilinmagan 3 olish imkonini beradi. Riman sferasining istalgan nuqtasida. Matematik Bernxard Rimann sharafiga nomlangan.

Ta'rif

Rieman differensial tenglamasi quyidagicha aniqlanadi

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

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

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

$\alpha +{\alpha }^{\prime }+\beta +{\beta }^{\prime }+\gamma +{\gamma }^{\prime }=1.$

Tenglama yechimlari

Riman tenglamasining yechimlari Riman doim P belgisi bilan yoziladi

$w=P\left\{\begin{array}{cccc}a& b& c& \\ \alpha & \beta & \gamma & z\\ {\alpha }^{\prime }& {\beta }^{\prime }& {\gamma }^{\prime }& \end{array}\right\}$

Odatiy gipergeometrik funktsiyani quyidagicha yozish mumkin bo'ladi

${}_2{F}_1(a,b;c;z)=P\left\{\begin{array}{cccc}0& \mathrm{\infty }& 1& \\ 0& a& 0& z\\ 1-c& b& c-a-b& \end{array}\right\}$

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

$P\left\{\begin{array}{cccc}a& b& c& \\ \alpha & \beta & \gamma & z\\ {\alpha }^{\prime }& {\beta }^{\prime }& {\gamma }^{\prime }& \end{array}\right\}={\left(\frac{z-a}{z-b}\right)}^{\alpha }{\left(\frac{z-c}{z-b}\right)}^{\gamma }P\left\{\begin{array}{cccc}0& \mathrm{\infty }& 1& \\ 0& \alpha +\beta +\gamma & 0& \frac{(z-a)(c-b)}{(z-b)(c-a)}\\ {\alpha }^{\prime }-\alpha & \alpha +{\beta }^{\prime }+\gamma & {\gamma }^{\prime }-\gamma & \end{array}\right\}$

shaklda tenglamaning yechimini yozish imkonini beradi

$w={\left(\frac{z-a}{z-b}\right)}^{\alpha }{\left(\frac{z-c}{z-b}\right)}^{\gamma }{}_2{F}_1(\alpha +\beta +\gamma ,\alpha +{\beta }^{\prime }+\gamma ;1+\alpha -{\alpha }^{\prime };\frac{(z-a)(c-b)}{(z-b)(c-a)})$

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

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

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

Keyingi tenglik

$P\left\{\begin{array}{cccc}a& b& c& \\ \alpha & \beta & \gamma & z\\ {\alpha }^{\prime }& {\beta }^{\prime }& {\gamma }^{\prime }& \end{array}\right\}=P\left\{\begin{array}{cccc}\eta & \zeta & \theta & \\ \alpha & \beta & \gamma & u\\ {\alpha }^{\prime }& {\beta }^{\prime }& {\gamma }^{\prime }& \end{array}\right\}$

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