Riman differentsial tenglamasi
Rieman differensial tenglamasi gipergeometrik tenglamani umumlashtirish boʻlib, u sizga muntazam yagona nuqtalar(ingl.)oʻzb. olish imkonini beradi. ) Riman sferasining istalgan nuqtasida. Matematik Bernxard Rimann sharafiga nomlangan.
Taʼrif
Rieman differensial tenglamasi quyidagicha aniqlanadi
d
2
w
d
z
2
+
[
1
−
α
−
α
′
z
−
a
+
1
−
β
−
β
′
z
−
b
+
1
−
γ
−
γ
′
z
−
c
]
d
w
d
z
{\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}}}
+
[
α
α
′
(
a
−
b
)
(
a
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)
z
−
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+
β
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]
w
(
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)
(
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(
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=
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 boʻladi. 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
{
a
b
c
α
β
γ
z
α
′
β
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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 boʻladi
2
F
1
(
a
,
b
;
c
;
z
)
=
P
{
0
∞
1
0
a
0
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1
−
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−
a
−
b
}
{\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 oʻziga xosliklarga boʻysunadi, ulardan biri ularni gipergeometrik funksiyalar nuqtai nazaridan umumlashtirish imkonini beradi. Yaʼni, u ifoda quyidagicha
P
{
a
b
c
α
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γ
z
α
′
β
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γ
′
}
=
(
z
−
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γ
P
{
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0
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+
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+
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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
−
a
z
−
b
)
α
(
z
−
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−
b
)
γ
2
F
1
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+
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+
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+
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1
+
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(
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)
{\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
p-funksiya Mebius oʻzgarishiga nisbatan oddiy simmetriyaga ega, yaʼni GL(2,
C ) yoki ekvivalenti bilan Riman sferasining konformal xaritasi deyiladi . Oʻzboshimchalik bilan tanlangan toʻrtta kompleks sonlar A, B, C va D shartni qondiradi
A
D
−
B
C
≠
0
{\displaystyle AD-BC\neq 0}
, nisbatlarini aniqlang.
u
=
A
z
+
B
C
z
+
D
and
η
=
A
a
+
B
C
a
+
D
{\displaystyle u={\frac {Az+B}{Cz+D}}\quad {\text{ and }}\quad \eta ={\frac {Aa+B}{Ca+D}}}
va
ζ
=
A
b
+
B
C
b
+
D
and
θ
=
A
c
+
B
C
c
+
D
.
{\displaystyle \zeta ={\frac {Ab+B}{Cb+D}}\quad {\text{ and }}\quad \theta ={\frac {Ac+B}{Cc+D}}.}
Keyingi tenglik
P
{
a
b
c
α
β
γ
z
α
′
β
′
γ
′
}
=
P
{
η
ζ
θ
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u
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β
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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
uz.wikipedia.org